\section{Walker conventions and the two Boolean state spaces}
\label{common:sec-walker-data}

An attachment state records which handles are present; a slide state
records which complete markings have changed after all handles are present.
Both use the same labelled handles, terminal surface, endpoint marking, and
Walker multiplication law.  Their Boolean indices nevertheless describe
different geometric operations.

We first fix the marked surface, its rational Lagrangian, and the Walker
multiplication convention.  The common relative extension then lets later
order and coherence arguments compare routes within each chart while keeping
the two kinds of state distinct.

\subsection{Marked Walker cylinders}

All manifolds and surfaces are oriented.  Composition and mapping-class
products are written right-factor first: in a product \(MN\), the factor
\(N\) is traversed first.  Let \((\Sigma,\lambda)\) be a marked oriented
surface with declared rational Lagrangian
\(\lambda\subset H_1(\Sigma;\mathbb Q)\).  For an orientation-preserving
marked diffeomorphism \(\phi\) of \(\Sigma\), write \(C(\phi,m)\) for its
Walker mapping cylinder of integer weight \(m\).  Our convention is
\begin{equation}
 C(\psi,n)C(\phi,m)
 =C\!\left(
   \psi\phi,\,
   n+m-\mu_\Sigma(\phi_*\lambda,\lambda,
                    \psi_*^{-1}\lambda)
 \right).
 \label{common:eq-walker-product}
\end{equation}
We use the extended-surface convention of
\cite{Walker1991} and
\cite[arXiv v3, \S3, eq.~(2)]{GilmerMasbaum2013}.
We use throughout the direct-sum and
repeated-pair identities
\begin{align}
 \mu(L_1\oplus L_1',L_2\oplus L_2',L_3\oplus L_3')
 &=\mu(L_1,L_2,L_3)+\mu(L_1',L_2',L_3'),
 \label{common:eq-maslov-direct-sum}\\
 \mu(L_1,L_2,L_3)&=0
 \quad\text{if two of the three entries coincide}.
 \label{common:eq-maslov-repeated-pair}
\end{align}
Every later calculation first displays the three Lagrangians in the order
forced by \eqref{common:eq-walker-product} and only then applies these two
identities.

Equality of Walker morphisms means literal equality in one marked Hom set,
with both endpoint markings and the integer weight retained.  The notation
\(M\cong N\) instead records a boundary-marking-preserving diffeomorphism of
the underlying cobordisms retaining the displayed weights.  If an underline
is used, \(\underline M\) forgets only the Walker integer and retains all
marking data.  The symbol \(M^\dagger\) is reserved for source orientation
reversal; it is neither a Hilbert-space adjoint nor the inverse of a slide
map.  For an oriented product-framed simple closed curve \(\gamma\), write
\(T_\gamma\) for the positive right Dehn twist in its specified collar,
extended by the identity outside that collar.

\subsection{The attachment-state Boolean chart}

For a positive integer \(n\), put \([n]=\{1,\ldots,n\}\).  Fix
\(g,k\geq1\), a connected marked outgoing surface
\((\Sigma_g^{\mathrm{out}},\lambda_g)\), and \(k\) pairwise-disjoint,
distinctly labelled disk sites \(d_1,\ldots,d_k\) with disjoint closures and
complete local markings.  A fixed-site \emph{attachment state} is a subset
\begin{equation}
 I\subseteq[k].
 \label{fixed:eq-attachment-state}
\end{equation}
Attaching the labelled one-holed torus at precisely the sites in \(I\) gives
the outgoing surface \(\Sigma_I^{\mathrm{out}}\).  Its marked homology and
declared Lagrangian split as
\begin{align}
 H_1(\Sigma_I^{\mathrm{out}};\mathbb Q)
 &=H_1(\Sigma_g^{\mathrm{out}};\mathbb Q)
   \oplus\bigoplus_{t\in I}
       \operatorname{span}_{\mathbb Q}\{[m_t],[\ell_t]\},
 \label{fixed:eq-attachment-homology}\\
 \lambda_I
 &=\lambda_g\oplus
   \bigoplus_{t\in I}\langle[m_t]\rangle_{\mathbb Q}.
 \label{fixed:eq-attachment-lagrangian}
\end{align}
The marking at site \(t\) includes its ordered attaching feet and joining
band, oriented co-core meridian \(m_t\), oriented product-framed longitude
\(\ell_t\), and separating collar, with
\([m_t]\cdot[\ell_t]=1\).  The old surface, every handle indexed by \(I\),
and every unused disk are labelled spectators for an attachment at a new
site.

Let
\begin{equation}
 Q_{g,k}^{\mathrm{out}}
 :=\operatorname{Mod}\!\left(
   \Sigma_g^{\mathrm{out}}
    \setminus\bigcup_{t=1}^k\operatorname{int}(d_t),\partial
  \right),
 \label{fixed:eq-relative-mapping-class-group}
\end{equation}
and let
\begin{equation}
 e_I:Q_{g,k}^{\mathrm{out}}\longrightarrow
 \Gamma(\Sigma_I^{\mathrm{out}})
 \label{fixed:eq-capping-extension}
\end{equation}
be capping followed by the identity extension over the attached labelled
tori.

\Needspace{4\baselineskip}
\begin{proposition}[Common Boolean Walker extension]
\label{fixed:prop-common-walker-extension}
The Walker two-cocycles pulled back along the maps \(e_I\),
\(I\subseteq[k]\), are equal.  Consequently there is one central extension
\(\widetilde Q_{g,k}^{\mathrm{out}}\) and, for every attachment state \(I\),
a homomorphism
\begin{equation}
 \widetilde e_I:\widetilde Q_{g,k}^{\mathrm{out}}\longrightarrow
 \widetilde\Gamma(\Sigma_I^{\mathrm{out}},\lambda_I),
 \qquad
 \widetilde e_I(q,n)=C(e_I(q),n).
 \label{fixed:eq-common-walker-extension}
\end{equation}
Thus one relative mapping class and its integer weight have compatible images
at every vertex of the attachment-state Boolean chart.
\end{proposition}

\begin{proof}
 Each \(e_I\) fixes the attached torus summands, so its pulled-back cocycle
splits into the old relative term and repeated meridian pairs.  The pairs
contribute zero.  The cocycle is therefore independent of the attachment
state, giving one central extension of the shared relative source.

Let \(f,h\in Q_{g,k}^{\mathrm{out}}\).  At the state \(I\), the correction
for the product of their extensions is
\[
 -\mu\!\left((e_I(f))_*\lambda_I,\lambda_I,
             (e_I(h))_*^{-1}\lambda_I\right).
\]
Both mapping classes are the identity on every attached labelled torus.
Using \eqref{fixed:eq-attachment-homology} and
\eqref{fixed:eq-attachment-lagrangian}, the displayed correction is
\begin{align*}
 &-\mu\!\left((e_\varnothing(f))_*\lambda_g,\lambda_g,
              (e_\varnothing(h))_*^{-1}\lambda_g\right)\\
 &\qquad
 -\sum_{t\in I}
   \mu(\langle[m_t]\rangle,\langle[m_t]\rangle,
       \langle[m_t]\rangle).
\end{align*}
Every term in the sum is zero by
\eqref{common:eq-maslov-repeated-pair}.  The pullback cocycle is therefore
the one at \(I=\varnothing\), independently of \(I\), and
\eqref{fixed:eq-common-walker-extension} respects
\eqref{common:eq-walker-product}.  The common source is the relative group
\(Q_{g,k}^{\mathrm{out}}\); this construction introduces no comparison
homomorphism between unrelated closed Walker groups.
\end{proof}

The terminal attachment state fixes the notation used by the slide layer:
\begin{equation}
 X:=\Sigma_\varnothing^{\mathrm{out}}=\Sigma_g^{\mathrm{out}},
 \qquad
 Y:=\Sigma_{[k]}^{\mathrm{out}}=\Sigma_{g+k}^{\mathrm{out}},
 \qquad
 \Lambda:=\lambda_{[k]}.
 \label{common:eq-terminal-interface-data}
\end{equation}
The terminal attachment trace is a compression body with complete marking
\(W:X\to Y\).  Its complete marking contains more than the symplectic
splitting: it retains the ordered feet and bands, oriented co-core disks,
embedded product-framed meridian--longitude rows, endpoint
parameterizations, signed cut faces and pairings, full product collars, dual
ribbons, and the interface and detector germs used in the slide construction.
The old marking also fixes the inclusion
\(H_1(X;\mathbb Q)\hookrightarrow H_1(Y;\mathbb Q)\), through which we regard
\(H_1(X;\mathbb Q)\) and \(\lambda_g\) as subspaces of
\(H_1(Y;\mathbb Q)\).  In particular,
\begin{align}
 H_1(Y;\mathbb Q)
 &=H_1(X;\mathbb Q)\oplus
   \bigoplus_{t=1}^k
    \operatorname{span}_{\mathbb Q}\{[m_t],[\ell_t]\},
 \label{common:eq-terminal-homology-splitting}\\
 \Lambda
 &=\lambda_g\oplus
   \bigoplus_{t=1}^k\langle[m_t]\rangle_{\mathbb Q}.
 \label{common:eq-terminal-lagrangian}
\end{align}

For an attachment order
\(\alpha=(\alpha_1,\ldots,\alpha_k)\in S_k\), define
\begin{equation}
 I_q^\alpha:=\{\alpha_1,\ldots,\alpha_q\},
 \qquad I_0^\alpha:=\varnothing.
 \label{fixed:eq-attachment-prefix}
\end{equation}
Independent forward and reverse attachment orders will be denoted
\(\alpha,\beta\in S_k\).  These symbols are never used for slide order.

\subsection{The completed slide-state Boolean cube}

The slide enhancement is available only after all \(k\) handles have been
attached.  It requires integers \(s\geq1\) and \(k\geq2s\), together with
the protected data specified in \cref{slide:sec-setup}.  A \emph{slide state}
is a subset
\begin{equation}
 A\subseteq[s].
 \label{slide:eq-slide-state}
\end{equation}
At every such state the underlying outgoing surface is the same surface
\(Y\); the state records which of the \(s\) prescribed positive slides have
been completed in the marking.  We write \(\mathfrak C_A\) for that completed
marking, \(p_A\) for its outgoing parameterization, and
\(m_t^A,\ell_t^A\subset Y\) for its actual oriented product-framed row.  The
associated Lagrangian will be
\begin{equation}
 \Lambda_A:=\lambda_g\oplus
   \bigoplus_{t=1}^k\langle[m_t^A]\rangle_{\mathbb Q}.
 \label{slide:eq:vertex-lagrangian}
\end{equation}
Only the reference identity \(\Lambda_\varnothing=\Lambda\) is immediate
from the old marking.  The equality \(\Lambda_A=\Lambda\) for arbitrary
\(A\) is a later theorem, proved from the signed integral edge formulas in
both directions; it is not part of the definition above.

For a slide order
\(\omega=(\omega_1,\ldots,\omega_s)\in S_s\), put
\begin{equation}
 A_q^\omega:=\{\omega_1,\ldots,\omega_q\},
 \qquad A_0^\omega:=\varnothing.
 \label{slide:eq-slide-prefix}
\end{equation}
Thus \(I=\varnothing\) is the genus-\(g\) attachment state, while
\(A=\varnothing\) is the reference marking on the already completed
genus-\(g+k\) boundary.  In particular,
\begin{equation}
 \Sigma_\varnothing^{\mathrm{out}}=X
 \quad\text{but}\quad
 (W,\mathfrak C_\varnothing)\text{ has outgoing boundary }Y.
 \label{common:eq-two-empty-states}
\end{equation}
No argument may identify these two empty states.

\subsection{Phases, labelled weights, and two reusable lemmas}

Fix phase assignments and independent label-attached weight vectors
\begin{equation}
 \epsilon,\eta\in\{0,1\}^k,
 \qquad
 \mathbf c,\mathbf r,\mathbf u^+,\mathbf u^-\in\mathbb Z^k.
 \label{common:eq-phase-weight-data}
\end{equation}
A coordinate \(u_t^+\) with \(\epsilon_t=0\), or \(u_t^-\) with
\(\eta_t=0\), is inactive and ignored.  Write
\begin{equation}
 \begin{aligned}
 c_\Sigma&:=\sum_{t=1}^k c_t,&
 r_\Sigma&:=\sum_{t=1}^k r_t,\\
 U^+(\epsilon)&:=\sum_{\epsilon_t=1}u_t^+,&
 U^-(\eta)&:=\sum_{\eta_t=1}u_t^-.
 \end{aligned}
 \label{common:eq-phase-weight-totals}
\end{equation}
The coordinates remain attached to their labels during every equality
replacement.  Only the displayed totals are intrinsic to a completed
morphism after the labelled construction has been performed.

The following lemma isolates the one phase-cylinder calculation common to
the two layers.  It does not concern handle arrows or slide edges.

\begin{lemma}[Disjoint phase cylinders]
\label{common:lem-disjoint-phase-cylinders}
Let \((Z,\Lambda_Z)\) be a marked surface with a symplectic splitting
\[
 H_1(Z;\mathbb Q)=H_{\mathrm{old}}\oplus
  \bigoplus_{t\in K}P_t,
 \qquad
 \Lambda_Z=\Lambda_{\mathrm{old}}\oplus
  \bigoplus_{t\in K}\langle[m_t]\rangle_{\mathbb Q},
\]
where \(P_t=\operatorname{span}_{\mathbb Q}\{[m_t],[\ell_t]\}\), and suppose
the corresponding one-holed-torus collars are pairwise disjoint.  Put
\[
 \sigma_t:=T_{m_t}T_{\ell_t}T_{m_t},
 \qquad
 \Theta_t(u):=C(\sigma_t,u).
\]
For distinct \(i,j\in K\) and \(u,v\in\mathbb Z\),
\begin{equation}
 \Theta_j(v)\Theta_i(u)
 =C(\sigma_j\sigma_i,u+v)
 =C(\sigma_i\sigma_j,u+v)
 =\Theta_i(u)\Theta_j(v).
 \label{common:eq-disjoint-phase-swap}
\end{equation}
More generally, every labelled product of distinct \(\Theta_t(u_t)\) is
independent of its order as a Walker morphism and has weight
\(\sum_tu_t\).
\end{lemma}

\begin{proof}
 The calculation separates over the disjoint symplectic planes.  The Walker
correction decomposes into the corresponding Maslov contributions, and each
active plane contributes a triple of meridian/longitude lines whose index is
zero.  The mapping classes commute because their collars are disjoint; the
longer-product statement then follows by adding one untouched plane at a time.

The action of \(\sigma_t\) sends the meridian line in \(P_t\) to the
longitude line and fixes every other summand.  In the first product in
\eqref{common:eq-disjoint-phase-swap}, the correction dictated by
\eqref{common:eq-walker-product} is
\[
 \mu_Z((\sigma_i)_*\Lambda_Z,\Lambda_Z,
       (\sigma_j)_*^{-1}\Lambda_Z).
\]
On \(P_i\) the three lines are
\((\langle\ell_i\rangle,\langle m_i\rangle,
  \langle m_i\rangle)\); on \(P_j\) they are
\((\langle m_j\rangle,\langle m_j\rangle,
  \langle\ell_j\rangle)\); and on every other summand all three entries
agree.  The correction is zero.  Reversing \(i\) and \(j\) gives the
independently ordered triples
\((m_i,m_i,\ell_i)\) and \((\ell_j,m_j,m_j)\), so that correction also
vanishes.  The mapping classes commute pointwise because their collars are
disjoint, proving \eqref{common:eq-disjoint-phase-swap}.

For a longer prefix, the new factor changes one previously untouched
symplectic plane.  Each old active plane contributes a triple of the form
\((\ell,m,m)\), the new plane contributes \((m,m,\ell)\), and every other
plane is constant.  Induction and
\eqref{common:eq-maslov-direct-sum}--
\eqref{common:eq-maslov-repeated-pair} prove the last assertion.
\end{proof}

The second reusable lemma contains only the combinatorics of literal
equalities.  All geometric and Walker hypotheses must be verified before it
is invoked.

\begin{lemma}[Adjacent-equality confluence]
\label{common:lem-adjacent-equality-confluence}
Let \(P\) be a finite set, let \(Z_B\) be an object for every \(B\subseteq P\),
and for \(p\notin B\) let
\(\Phi_{p\mid B}:Z_B\to Z_{B\cup\{p\}}\) be a typed arrow.  Suppose that for
every distinct \(p,q\notin B\) one has the literal equality
\begin{equation}
 \Phi_{q\mid B\cup\{p\}}\Phi_{p\mid B}
 =\Phi_{p\mid B\cup\{q\}}\Phi_{q\mid B}.
 \label{common:eq-abstract-literal-square}
\end{equation}
Then the total composite is independent of the order of \(P\).  Adjacent
equality replacements close around the involutive, distant, and
three-term Coxeter-shaped diagrams.  These diagrams retain no comparison
arrow and define no permutation action.  The same assertion holds in the
opposite category.
\end{lemma}

\begin{proof}
Every two orders of \(P\) are related by adjacent transpositions.  At an
adjacent transposition, the preceding entries determine one subset \(B\),
and the affected two-factor block is exactly one instance of
\eqref{common:eq-abstract-literal-square}; all exterior factors have the same
typed endpoints.  Repeated substitution proves order independence.

Applying the same equality twice gives the involutive loop.  Replacements in
disjoint pairs of positions commute because they produce the same labelled
factors with the same preceding subsets.  For three consecutive labels, the
two sequences of three adjacent replacements begin with the same expression
and end with the same reversed expression; every intermediate edge is an
instance of \eqref{common:eq-abstract-literal-square}.  Transitivity of
literal equality closes the two routes.  No new arrow has been constructed.
Passing to the opposite category proves the final statement.
\end{proof}