\section{Strict transport of cube edges, phases, and traces}
\label{trfb1odmscoc:sec-strict-operators-phases-traces}

The fixed comparison identifies the finite skein fibre with the TQFT state
space, so a geometric edge now acts on the skein fibre by conjugation.
Retaining the Walker integer records its central scalar, and retaining the
composition order lets the phase changes and forward/reverse traces obey the
same identities as their source cobordisms.

Fix all data of Sections~\ref{sec:standing-inputs} and
\ref{trfb1odmscoc:sec-root-reduced-crosswalk}.  In particular, \(K\geq1\),
\(d=g+k\geq3\), and every surface, Lagrangian, marking, cylinder, phase,
and completed trace is the corresponding fixed object in the supplied
geometric cube input.  Write

\[
 Q:=Q_{K,d}:\mathcal M_{K,d}\xrightarrow{\sim}\mathcal V^+_{K,d}
\]

for the single old-vertex comparison isomorphism of
Proposition~\ref{trfb1odmscoc:prop-q-iso}.  It is never replaced by a
vertex-dependent map.  Products of source cobordisms and of their images
are written right-factor first.  Every source equality used below is a
literal Walker equality supplied by the geometric input; none is inferred from a target
operator equality.

First fix the strict scalar-extended Walker representation and the one
comparison map that identifies the finite source fibre.  Next transport each
weight-zero slide edge, including its twist-word correction, without replacing
the geometric edge by a skein presentation.  Then add the completed endpoint
phases and their central weights and edge covariance.  Finally factor the
forward and reverse traces separately, since their high boundaries have
opposite categorical roles.  The arguments below follow this order:
strict functoriality first, supplied source equalities second, and matrix or
scalar consequences only after the typed transport is complete.

For an oriented product-framed simple closed curve \(\alpha\subset Y\), let
\(T_\alpha\) denote the positive (right) Dehn twist supported in its specified
one-holed-torus collar.  For a signed twist word
\(\mathfrak w=\alpha_1^{\varepsilon_1}\cdots
\alpha_q^{\varepsilon_q}\), write
\[
 \operatorname{Tw}(\mathfrak w)
 :=T_{\alpha_1}^{\varepsilon_1}\cdots
   T_{\alpha_q}^{\varepsilon_q}.
\]

\subsection{The strict scalar-extended Walker representation}

Let \(Z_K=E_K\otimes_{k_{p,K}}Z_p\).  Scalar extension preserves the
composition and identity axioms of the fixed-level extended TQFT, so for
composable extended cobordisms

\begin{equation}
 Z_K(M_2M_1)=Z_K(M_2)Z_K(M_1),
 \qquad
 Z_K(C_Y(\operatorname{id}_Y,0))=\operatorname{id}.
 \label{trfb1odmscoc:eq-strict-Z}
\end{equation}

The Walker weight of the composite on the left is already the
Maslov-corrected weight.  Thus this is a strict functor on extended
cobordisms, not the unextended projective mapping-class action.

\begin{proposition}[Strict representation in the fixed comparison model]
\label{trfb1odmscoc:prop-strict-representation}
For every high-surface extended cylinder \(G\), define

\begin{equation}
 \rho^Q_{K,d}(G):=Q^{-1}Z_K(G)Q.
 \label{trfb1odmscoc:eq-rhoQ-definition}
\end{equation}

Then

\begin{align}
 \rho^Q_{K,d}(G_2G_1)
 &=\rho^Q_{K,d}(G_2)\rho^Q_{K,d}(G_1),
 \label{trfb1odmscoc:eq-rhoQ-product}\\
 \rho^Q_{K,d}(G^{-1})
 &=\rho^Q_{K,d}(G)^{-1}.
 \label{trfb1odmscoc:eq-rhoQ-inverse}
\end{align}

If \(W_Y=C_Y(\operatorname{id}_Y,1)\), then for every \(n\in\mathbb Z\),

\begin{equation}
 \rho^Q_{K,d}(W_Y^nG)
 =\rho^Q_{K,d}(GW_Y^n)
 =\kappa_K^n\rho^Q_{K,d}(G).
 \label{trfb1odmscoc:eq-rhoQ-central}
\end{equation}

Finally, a supplied literal source equality

\begin{equation}
 G_r\cdots G_1=G'_q\cdots G'_1
 \label{trfb1odmscoc:eq-source-equality-template}
\end{equation}

implies the literal operator equality

\begin{equation}
 \rho^Q_{K,d}(G_r)\cdots\rho^Q_{K,d}(G_1)
 =\rho^Q_{K,d}(G'_q)\cdots\rho^Q_{K,d}(G'_1).
 \label{trfb1odmscoc:eq-one-way-shadow}
\end{equation}

There is no converse implication.
\end{proposition}

\begin{proof}
Strict functoriality preserves the composition of extended cobordisms.
When the fixed comparison map changes the source fibre, its adjacent inverse
factors cancel in each product.  The central cylinder records the weight
character, so this conjugation transports a supplied Walker equality with
its integer weight intact.

Insert \(QQ^{-1}=\operatorname{id}_{\mathcal V^+_{K,d}}\) between the
two factors in \eqref{trfb1odmscoc:eq-strict-Z}:

\[
 Q^{-1}Z_K(G_2G_1)Q
 =Q^{-1}Z_K(G_2)QQ^{-1}Z_K(G_1)Q.
\]

This proves \eqref{trfb1odmscoc:eq-rhoQ-product}.  Applying it to
\(G^{-1}G=GG^{-1}=C_Y(\operatorname{id}_Y,0)\) proves both sides of
\eqref{trfb1odmscoc:eq-rhoQ-inverse}.

Increasing the Walker weight by one multiplies \(Z_p\), and hence its
scalar extension, by \(\kappa_K\).  The same statement for arbitrary
positive and negative \(n\), together with centrality of \(W_Y\), gives
\eqref{trfb1odmscoc:eq-rhoQ-central}.  Repeated use of
\eqref{trfb1odmscoc:eq-rhoQ-product} on the two sides of the already
supplied equality \eqref{trfb1odmscoc:eq-source-equality-template}
proves \eqref{trfb1odmscoc:eq-one-way-shadow}.  This last argument only
applies a function to a source equality and therefore creates no
target-to-source inference.
\end{proof}

We use the right-twist convention of
\cite[Section~4, Theorem~4.2 and Definition~4.3; Section~11, Theorem~11.2]{GilmerMasbaum2013}.
For a word \(\mathfrak w\), its surgery link has the curve
\(\alpha_i\) with framing \(-\varepsilon_i\) relative to the surface,
ordered with \(\alpha_q\) nearest \(0\times Y\) and
\(\alpha_1\) nearest \(1\times Y\).  Cabling every component by
the source's \(\omega\) gives \(s(\mathfrak w)\).
The corrected link is formed in the source's Heegaard embedding,
with \(\Lambda\) the first handlebody disk kernel, by adjoining the
zero-framed unlink of pushed-off meridians in the complement.
Here \(L^0_\Lambda(\mathfrak w)\) denotes its linking matrix;
the source uses that notation for the framed link itself.

\begin{proposition}[Exact word-signature correction]
\label{trfb1odmscoc:prop-word-correction}
Let \(f\in\Gamma(Y)\), \(n\in\mathbb Z\), and let

\[
 \mathfrak w=\alpha_1^{\varepsilon_1}\cdots
 \alpha_q^{\varepsilon_q},
 \qquad \operatorname{Tw}(\mathfrak w)=f.
\]

For the prescribed surgery skein \(s(\mathfrak w)\) and corrected
linking-matrix signature

\[
 n^0_\Lambda(\mathfrak w)
 :=\operatorname{sign}\bigl(L^0_\Lambda(\mathfrak w)\bigr),
\]

one has

\begin{equation}
 Z_K(C_Y(f,n))
 =\kappa_K^{\,n-n^0_\Lambda(\mathfrak w)}
  Z_K(I\times Y,s(\mathfrak w)).
 \label{trfb1odmscoc:eq-word-correction}
\end{equation}

In particular,

\begin{equation}
 Z_K(C_Y(f,0))
 =\kappa_K^{-n^0_\Lambda(\mathfrak w)}
  Z_K(I\times Y,s(\mathfrak w)).
 \label{trfb1odmscoc:eq-weight-zero-word}
\end{equation}

The corrected expression is independent of the chosen Dehn-twist word.
\end{proposition}

\begin{proof}
 The twist word is used as a presentation of an already fixed extended
cylinder.  Surgery supplies its corrected Walker weight, and changing the
weight to the requested integer contributes the central character.  Word
independence follows only because both corrected expressions represent the
same typed extended cylinder, not because a bare skein word has been declared
canonical.

Gilmer--Masbaum's surgery and TQFT word formulas
\cite[Theorems~4.2 and~11.2, equation~(21)]{GilmerMasbaum2013} say that extended surgery on the
weight-zero identity cylinder along the framed link for \(\mathfrak w\)
produces \(C_Y(f,n^0_\Lambda(\mathfrak w))\).  The surgery axiom therefore
identifies the bare surgery operator with
\[
 Z_K(C_Y(f,n^0_\Lambda(\mathfrak w))).
\]
Changing that Walker weight to
\(n\) multiplies by
\(\kappa_K^{n-n^0_\Lambda(\mathfrak w)}\), which proves
\eqref{trfb1odmscoc:eq-word-correction}.  Setting \(n=0\) gives
\eqref{trfb1odmscoc:eq-weight-zero-word}; the minus sign is forced by
the direction of the weight change.  If two words represent \(f\), both
corrected expressions equal the same already typed extended cylinder
\(C_Y(f,n)\), so word independence is a consequence rather than a bare
skein relation.
\end{proof}

\subsection{Weight-zero slide edges}

For \(A\subseteq[s]\) and \(a\notin A\), set

\begin{equation}
 T^K_{a\mid A}
 :=\rho^Q_{K,d}(H_{a\mid A})
 =Q^{-1}Z_K(C_Y(h_{a\mid A},0))Q.
 \label{trfb1odmscoc:eq-edge-definition}
\end{equation}

\begin{proposition}[Typed edge inverse and word formula]
\label{trfb1odmscoc:prop-edge-operator}
Every \(T^K_{a\mid A}\) is an \(E_K\)-linear automorphism of
\(\mathcal M_{K,d}\), with

\begin{equation}
 (T^K_{a\mid A})^{-1}
 =Q^{-1}Z_K(C_Y(h_{a\mid A}^{-1},0))Q.
 \label{trfb1odmscoc:eq-edge-inverse}
\end{equation}

For every Dehn-twist word \(\mathfrak w_{a\mid A}\) representing
\(h_{a\mid A}\),

\begin{equation}
 T^K_{a\mid A}
 =Q^{-1}\!\left(
   \kappa_K^{-n^0_\Lambda(\mathfrak w_{a\mid A})}
   Z_K(I\times Y,s(\mathfrak w_{a\mid A}))
  \right)Q.
 \label{trfb1odmscoc:eq-edge-word}
\end{equation}
\end{proposition}

\begin{proof}
 Invertibility is established in the source category first: each product with
the proposed inverse has identity underlying map and zero Maslov correction because the
edge preserves the declared Lagrangian in both directions.  The twist-word
formula is then just the one-way word correction conjugated by the same old
comparison map, so it does not introduce a new edge or a new comparison map.

The geometric cube theorem supplies both
\((h_{a\mid A})_*\Lambda=\Lambda\) and
\((h_{a\mid A}^{-1})_*\Lambda=\Lambda\).  Consequently both products of
the proposed inverse cylinder with \(H_{a\mid A}\) have underlying map
the identity, and every Lagrangian in the ordered Walker seam is
\(\Lambda\).  Their correction is therefore
\(\mu_Y(\Lambda,\Lambda,\Lambda)=0\), so

\[
 C_Y(h_{a\mid A}^{-1},0)C_Y(h_{a\mid A},0)
 =C_Y(\operatorname{id}_Y,0)
 =C_Y(h_{a\mid A},0)C_Y(h_{a\mid A}^{-1},0).
\]

Apply \eqref{trfb1odmscoc:eq-rhoQ-inverse} to obtain
\eqref{trfb1odmscoc:eq-edge-inverse}.  Equation
\eqref{trfb1odmscoc:eq-edge-word} is
\eqref{trfb1odmscoc:eq-weight-zero-word} conjugated by the one fixed
map \(Q\).  It computes the image of the already constructed geometric
edge; it does not manufacture a slide from a skein word.
\end{proof}

\subsection{Completed endpoint phases}

At each completed vertex \(A\subseteq[s]\), let
\(\bar m_t^A,\bar\ell_t^A\) be the actual oriented product-framed row on
\(\partial_+(W,\mathfrak C_A)\), and let
\[
 m_t^A:=p_A(\bar m_t^A),\qquad
 \ell_t^A:=p_A(\bar\ell_t^A)\subset Y
\]
be its oriented counterpart in the fixed abstract surface, with its actual
one-holed-torus support collar.  For \(t\in[k]\) and \(u\in\mathbb Z\), retain
the geometric input definitions

\[
 \sigma_{t,A}=T_{m_t^A}T_{\ell_t^A}T_{m_t^A},
 \qquad
 \Theta_{t,A}(u)=C_Y(\sigma_{t,A},u),
\]

and define

\begin{equation}
 P^K_{t,A}(u):=\rho^Q_{K,d}(\Theta_{t,A}(u)).
 \label{trfb1odmscoc:eq-elementary-phase}
\end{equation}

Fix \(\mathbf u^+,\mathbf u^-\in\mathbb Z^k\).  For
\(\epsilon\in\{0,1\}^k\), let
\(I_\epsilon=\{t:\epsilon_t=1\}\) and
\(U^+(\epsilon)=\sum_{t\in I_\epsilon}u_t^+\).  If
\(\varrho=(\varrho_1,\ldots,\varrho_q)\) orders \(I_\epsilon\), put

\begin{equation}
 P^K_{A,\varrho}(\epsilon,\mathbf u^+)
 :=P^K_{\varrho_q,A}(u^+_{\varrho_q})\cdots
   P^K_{\varrho_1,A}(u^+_{\varrho_1}).
 \label{trfb1odmscoc:eq-ordered-phase-word}
\end{equation}

The corresponding common source data are

\begin{equation}
 \sigma_A^\epsilon:=\prod_{t\in I_\epsilon}\sigma_{t,A},
 \qquad
 S_A^\epsilon(\mathbf u^+)
 :=C_Y(\sigma_A^\epsilon,U^+(\epsilon)),
 \label{trfb1odmscoc:eq-common-source-phase}
\end{equation}

where the mapping-class product is the supplied
order-independent product.  For the reverse assignment, set
\(I_\eta=\{t:\eta_t=1\}\),
\(U^-(\eta)=\sum_{t\in I_\eta}u_t^-\), and define
\(S_A^\eta(\mathbf u^-)\) by the identical construction.

\begin{proposition}[Common phase, central weight, and edge conjugacy]
\label{trfb1odmscoc:prop-phases}
For every ordering \(\varrho\),

\begin{equation}
 P^K_{A,\varrho}(\epsilon,\mathbf u^+)
 =P^K_A(\epsilon,\mathbf u^+)
 :=\rho^Q_{K,d}(S_A^\epsilon(\mathbf u^+)).
 \label{trfb1odmscoc:eq-common-phase}
\end{equation}

This is the identity when \(I_\epsilon=\varnothing\).  For arbitrary
integer weights,

\begin{align}
 P^K_{t,A}(u)&=\kappa_K^uP^K_{t,A}(0),
 \label{trfb1odmscoc:eq-elementary-phase-weight}\\
 P^K_A(\epsilon,\mathbf u^+)
 &=\kappa_K^{U^+(\epsilon)}P^K_A(\epsilon,\mathbf0).
 \label{trfb1odmscoc:eq-common-phase-weight}
\end{align}

If \(A'=A\cup\{a\}\), then

\begin{equation}
 T^K_{a\mid A}P^K_A(\epsilon,\mathbf u^+)
 =P^K_{A'}(\epsilon,\mathbf u^+)T^K_{a\mid A}.
 \label{trfb1odmscoc:eq-phase-conjugacy}
\end{equation}

The same statements hold for the reverse assignment
\(\eta\in\{0,1\}^k\) and weights \(\mathbf u^-\).
\end{proposition}

\paragraph{Transporting phase data.}
Phase transport has two independent parts.  Source-level disjoint support
gives a common ordered cylinder and source-level covariance moves that cylinder
across an edge; the strict representation then transports both facts to the
finite fibre.  Central weights are added through the one central cylinder, so
the labels on the phase parameters remain visible throughout the argument.

\begin{proof}
The geometric cube theorem first supplies, at the source, the literal equality between every
ordered product of the cylinders \(\Theta_{t,A}(u_t^+)\) and the common
cylinder

\[
 S_A^\epsilon(\mathbf u^+)
 =C_Y(\sigma_A^\epsilon,U^+(\epsilon)).
\]

Apply the one-way shadow
\eqref{trfb1odmscoc:eq-one-way-shadow}; this proves
\eqref{trfb1odmscoc:eq-common-phase} while keeping each label \(t\)
attached to its integer \(u_t^+\).  The empty source word is the Walker
identity, so its image is the identity.

The source cylinders satisfy
\[
 \Theta_{t,A}(u)=W_Y^u\Theta_{t,A}(0),
 \qquad
 S_A^\epsilon(\mathbf u^+)
 =W_Y^{U^+(\epsilon)}S_A^\epsilon(\mathbf0).
\]
Equation \eqref{trfb1odmscoc:eq-rhoQ-central} proves
\eqref{trfb1odmscoc:eq-elementary-phase-weight} and
\eqref{trfb1odmscoc:eq-common-phase-weight}, including negative and zero
weights.

Finally the geometric cube theorem supplies the literal Walker equality

\[
 H_{a\mid A}S_A^\epsilon(\mathbf u^+)
 =S_{A'}^\epsilon(\mathbf u^+)H_{a\mid A}.
\]

Its image under the same \(Q\) on every factor is exactly
\eqref{trfb1odmscoc:eq-phase-conjugacy}.  The reverse phase statement is
identical after replacing \((\epsilon,\mathbf u^+)\) by
\((\eta,\mathbf u^-)\).
\end{proof}

\subsection{Forward and reverse trace maps}

For label-indexed vectors \(\mathbf c,\mathbf r\in\mathbb Z^k\), write

\[
 c_\Sigma=\sum_{t=1}^kc_t,
 \qquad
 r_\Sigma=\sum_{t=1}^kr_t.
\]

For the supplied completed source cobordisms
\(J_A^\epsilon(\mathbf c,\mathbf u^+):X\to Y\) and\linebreak[4]
\(R_A^\eta(\mathbf r,\mathbf u^-):Y\to X\), define

\begin{align}
 \mathsf J^K_A(\epsilon;\mathbf c,\mathbf u^+)
 &:=Q^{-1}Z_K(J_A^\epsilon(\mathbf c,\mathbf u^+)):
 \mathcal V^-_{K,g}\longrightarrow\mathcal M_{K,d},
 \label{trfb1odmscoc:eq-J-definition}\\
 \mathsf R^K_A(\eta;\mathbf r,\mathbf u^-)
 &:=Z_K(R_A^\eta(\mathbf r,\mathbf u^-))Q:
 \mathcal M_{K,d}\longrightarrow\mathcal V^-_{K,g}.
 \label{trfb1odmscoc:eq-R-definition}
\end{align}

\begin{proposition}[Phase factorization, weights, and both covariances]
\label{trfb1odmscoc:prop-trace-maps}
The completed maps factor as

\begin{align}
 \mathsf J^K_A(\epsilon;\mathbf c,\mathbf u^+)
 &=P^K_A(\epsilon,\mathbf u^+)
   \mathsf J^K_A(\mathbf0;\mathbf c,\mathbf0),
 \label{trfb1odmscoc:eq-J-phase-factor}\\
 \mathsf R^K_A(\eta;\mathbf r,\mathbf u^-)
 &=\mathsf R^K_A(\mathbf0;\mathbf r,\mathbf0)
   P^K_A(\eta,\mathbf u^-).
 \label{trfb1odmscoc:eq-R-phase-factor}
\end{align}

Their exact central-weight dependence is

\begin{align}
 \mathsf J^K_A(\epsilon;\mathbf c,\mathbf u^+)
 &=\kappa_K^{c_\Sigma+U^+(\epsilon)}
   \mathsf J^K_A(\epsilon;\mathbf0,\mathbf0),
 \label{trfb1odmscoc:eq-J-weight}\\
 \mathsf R^K_A(\eta;\mathbf r,\mathbf u^-)
 &=\kappa_K^{r_\Sigma+U^-(\eta)}
   \mathsf R^K_A(\eta;\mathbf0,\mathbf0).
 \label{trfb1odmscoc:eq-R-weight}
\end{align}

Here only the integer weights are zeroed on the right; the phase
assignment remains fixed.  If \(A'=A\cup\{a\}\), then

\begin{align}
 T^K_{a\mid A}\,
 \mathsf J^K_A(\epsilon;\mathbf c,\mathbf u^+)
 &=\mathsf J^K_{A'}(\epsilon;\mathbf c,\mathbf u^+),
 \label{trfb1odmscoc:eq-J-covariance}\\
 \mathsf R^K_{A'}(\eta;\mathbf r,\mathbf u^-)
 T^K_{a\mid A}
 &=\mathsf R^K_A(\eta;\mathbf r,\mathbf u^-).
 \label{trfb1odmscoc:eq-R-covariance}
\end{align}
\end{proposition}

\paragraph{Forward and reverse traces.}
The forward and reverse traces are factored separately because their high
boundary has opposite categorical roles.  Strict functoriality supplies the
two factorizations, the central character supplies their exact weights, and
the already established phase and edge covariances supply the two transport
identities.  No identification of the forward and reverse maps is made.

\begin{proof}
The supplied endpoint definitions are literal source factorizations

\[
 J_A^\epsilon(\mathbf c,\mathbf u^+)
 =S_A^\epsilon(\mathbf u^+)
  J_A^{\mathbf0}(\mathbf c,\mathbf0),
\]

and

\[
 R_A^\eta(\mathbf r,\mathbf u^-)
 =R_A^{\mathbf0}(\mathbf r,\mathbf0)
  S_A^\eta(\mathbf u^-).
\]

Apply strict functoriality and insert \(QQ^{-1}\) on the high side.
The resulting two typed compositions are precisely
\eqref{trfb1odmscoc:eq-J-phase-factor} and
\eqref{trfb1odmscoc:eq-R-phase-factor}.  Notice that their orders differ
because the high boundary is the target of \(J\) but the source of \(R\).

The geometric cube theorem also identifies the completed forward and reverse weights as
\(c_\Sigma+U^+(\epsilon)\) and \(r_\Sigma+U^-(\eta)\), respectively, relative to the
same phase assignments with every integer weight zero.  Applying the
central character \eqref{trfb1odmscoc:eq-rhoQ-central} on the high side,
or equivalently the fixed-level weight axiom before inserting \(Q\), gives
\eqref{trfb1odmscoc:eq-J-weight} and
\eqref{trfb1odmscoc:eq-R-weight}.

For covariance, the supplied source equalities are independently typed as

\[
 H_{a\mid A}J_A^\epsilon(\mathbf c,\mathbf u^+)
 =J_{A'}^\epsilon(\mathbf c,\mathbf u^+):X\longrightarrow Y,
\]

and

\[
 R_{A'}^\eta(\mathbf r,\mathbf u^-)
 H_{a\mid A}
 =R_A^\eta(\mathbf r,\mathbf u^-):Y\longrightarrow X.
\]

For the first, apply \(Q^{-1}Z_K(-)\) and insert \(QQ^{-1}\) between the
two source factors.  For the second, apply \(Z_K(-)Q\) and make the same
insertion.  These calculations give
\eqref{trfb1odmscoc:eq-J-covariance} and
\eqref{trfb1odmscoc:eq-R-covariance} separately.  No adjoint, dagger, or
unitarity statement connects the two arguments.
\end{proof}

The results of this section establish strict functoriality, the corrected
word formula, the edge and phase operators, and the two trace families.
They use the old comparison map only through
Proposition~\ref{trfb1odmscoc:prop-q-iso}; they do not prove a source
relation, introduce a slid-vertex basis, construct a low-side gauge, or
assert any finite-level dagger structure.