\section{Main theorem and construction order}
\label{sec:main-theorem}

The edge maps must fit together at their endpoints as marked maps.  Choosing
each slide representative separately would not ensure that two incident
edges use the same disk charts, collars, or spectator data.  We therefore
choose all five vertex systems before the endpoint charts and edge maps.
The theorem constructs one such marked system and compares its two routes.

Whenever a signed bicollar appears below, its negative normal coordinate is on
the local side and its positive normal coordinate is on the spectator side.
The later collar construction gives the corresponding product coordinates.

Full-normal pairing identities allow the cut maps to descend and extend to
global maps.  Transport of the embedded meridian and dual-longitude rows
then gives their homology blocks and the common preserved Lagrangian.  The
relative isotopy compares the two composites with their complete protected
collar fixed, and the preserved Lagrangian makes the Walker seam corrections
vanish.  Each step concerns the same selected maps and endpoint data.

Let
\[
 \cT=(g,N,W:X\longrightarrow Y_{\mathrm{geom}},i,j,k,
 \mathfrak C_{\mathrm{ext}},\lambda_g,\cL_{ijk})
 \tag{4.1}\label{eq:input-tuple}
\]
be a supplied protected-sector tuple in the sense of
\cref{sec:protected-sectors}.  Thus \(g\geq 1\), \(N\geq 3\), the indices
\(i,j,k\in\{1,\ldots,N\}\) are pairwise distinct, and
\(\cL_{ijk}\) contains the complete protected local geometry rather than
merely its homology classes.  Recall the vertex and edge sets
\[
 V=\{0,B,A,AC,\starv\},\qquad
 E_5=\{0B,B\starv,0A,AAC,AC\starv\}.
 \tag{4.2}\label{eq:pentagon-sets-main}
\]

\Needspace{4\baselineskip}
\begin{theorem}[Positive handleslide pentagon with complete markings]
\label{thm:complete-marked-pentagon}
For every tuple \(\cT\) in \eqref{eq:input-tuple}, there is a coordinated
system with the following properties.

\begin{enumerate}[label=\textup{(\roman*)},leftmargin=2.5em]
\item \emph{One vertex-first package.}
There is first one simultaneous choice
\[
 \mathbf V=\left(
  \mathfrak C_v,K_v,\cQ_v,q_v^P,q_v,q_v^{(3)},
  \rho_v^{(3)},\rho_v,p_v,\bar z_v,
  z_v:D^2\times[0,u_0]\longrightarrow K_v
 \right)_{v\in V}
 \tag{4.3}\label{eq:vertex-package}
\]
with one positive, vertex-only depth \(u_0\).  Every edge incident to a
vertex uses the literal corresponding member of \(\mathbf V\); there are no
edge-indexed replacements.  On the boundary of the disk collar,
\[
 p_v\rho_v z_v(x,0)=\bar z_v(x).
 \tag{4.4}\label{eq:correct-rc0}
\]
The trace map \(\rho_v\) in \eqref{eq:correct-rc0} is essential for the
types to match.

\item \emph{Ten endpoint atlases followed by five edge triples.}
Only after \(\mathbf V\) has been fixed are there a common depth
\(0<u_{\mathrm{cmp}}<u_0\), ten incidence-labelled spectator atlases, and
five coherent triples.  The incidence set is
\begin{align*}
 \mathcal I_{10}=\{&(0B,0),(0B,B),(B\starv,B),(B\starv,\starv),
 (0A,0),(0A,A),\\
 &(AAC,A),(AAC,AC),(AC\starv,AC),(AC\starv,\starv)\}.
\end{align*}
For each \((e,u)\in\mathcal I_{10}\) there is an atlas \(\Xi^z_{e,u}\).  Equal
atlas values do not erase incidence labels.  For every edge \(e:v\to w\)
there is then one triple
\[
 \Fcut e:K_v\longrightarrow K_w,\qquad
 \Fquot e:U_{ijk,v}\longrightarrow U_{ijk,w},\qquad
 \Fglob e:(W,\mathfrak C_v)\longrightarrow(W,\mathfrak C_w),
 \tag{4.5}\label{eq:three-map-triple}
\]
of orientation-preserving diffeomorphisms.  The first is the cut-ball map,
the second its quotient descent, and the third its global extension.  They
obey
\begin{align}
 \Fcut e z_v&=z_w
 &&\text{on }D^2\times[0,u_{\mathrm{cmp}}],
 \tag{4.6}\label{eq:z-comparison-main}\\
 \Fquot e\rho_v^{(3)}&=\rho_w^{(3)}\Fcut e,
 \tag{4.7}\label{eq:descent-conjugacy-main}\\
 \Fglob e|_{U_{ijk,v}}&=\Fquot e,
 &\Fglob e|_{W\setminus\Int U_{ijk,v}}&=\id,
 \tag{4.8}\label{eq:global-extension-main}\\
 (\Fglob e|_{\partial_+W})\rho_v
 &=\rho_w(\Fcut e|_{\partial K_v}).
 \tag{4.9}\label{eq:boundary-conjugacy-main}
\end{align}
The map \(\Fquot e\) is the unique map induced by \(\Fcut e\), and its
smooth inverse is induced by \((\Fcut e)^{-1}\).

\item \emph{All-jet stabilization and finite compatibility.}
For each edge, the local and spectator pieces meet along an internal disk
\(E_e\), which is not the global attachment disk \(E\).  Their extensions
agree to every order on an open signed bicollar of \(E_e\).  All fifteen
full-normal pairing conjugacies, in both directed forms, hold before
quotient descent.  After descent and global extension, the same three-stage
map system satisfies the complete stabilization and finite-compatibility
conditions.

More precisely, the finite compatibility package has 45 ordered group
headers and 185 typed records, with per-edge group sizes
\[
 (7,6,2,12,1,1,3,4,1).
 \tag{4.10}\label{eq:pc5-counts-main}
\]
At \(N=3\) the five unused-handle records remain in this typed census but
their domains are empty.  Write
\(\eps_{\mathrm{pair}}^{\mathrm{src}}>0\) for the common full-normal
pairing depth obtained during the vertex construction.  The package then
defines the common subordinate depths
\[
 u_{\mathrm{pair}}
 =\tfrac12\min\{u_{\mathrm{cmp}},\eps_{\mathrm{pair}}^{\mathrm{src}}\},
 \qquad
 \eps_{\mathrm{PC5,work}}=\tfrac12u_{\mathrm{cmp}},
 \qquad
 \eps_{\mathrm{PC5,pair}}=\tfrac12u_{\mathrm{pair}}.
 \tag{4.11}\label{eq:fresh-depths-main}
\]
In particular, no source depth is silently reused as a later compatibility
depth.

\item \emph{Domains and roles of the three map stages.}
The three maps in \eqref{eq:three-map-triple} form one system
\[
 \Fcut e\longmapsto\Fquot e\longmapsto\Fglob e.
 \tag{4.12}\label{eq:map-registry-main}
\]
Cut-level face, ribbon, spoke, pairing, and internal-interface assertions
belong to \(\Fcut e\).  Quotient descent belongs to \(\Fquot e\).  The
 two-sided global-\(E\) collar, the old region, and unused handles belong only
to \(\Fglob e\), after descent and exterior extension.  With these domains fixed,
the two cut routes use the same vertex data and their discrepancy is the
identity on one complete protected boundary collar.

\item \emph{Homology and the common Lagrangian.}
Put
\[
 U=\Span_{\Q}\{m_i,m_j,m_k\},\qquad
 \Lambda=\lambda_g^Y\oplus
 \bigoplus_{t\notin\{i,j,k\}}\Q[m_t]\oplus U.
 \tag{4.13}\label{eq:common-lagrangian-main}
\]
For \(e:v\to w\), let
\[
 h_e=p_w(\Fglob e|_{\partial_+W})p_v^{-1}:Y\longrightarrow Y.
 \tag{4.14}\label{eq:outgoing-map-main}
\]
The same global maps \(\Fglob e\) transport the embedded meridian and
product-framed dual-longitude rows.  Relative to the resulting splitting,
\[
 h_{e*}=I_{\mathrm{ext}}\oplus
 \begin{pmatrix}M_e&0_3\\0_3&M_e^{-T}\end{pmatrix}.
 \tag{4.15}\label{eq:homology-block-main}
\]
Both zero mixed blocks are geometric conclusions; neither is inferred from
symplecticity.  The actual maps and their actual inverses satisfy
\[
 h_{e*}\Lambda=\Lambda=h_{e*}^{-1}\Lambda.
 \tag{4.16}\label{eq:lag-main}
\]
When \(N=3\), the unused-handle sum in \eqref{eq:common-lagrangian-main}
is empty.

\item \emph{Relative pentagon and mapping classes.}
In right-factor-first order, set
\[
 F_L=F_{B\starv}\compose F_{0B},\qquad
 F_R=F_{AC\starv}\compose F_{AAC}\compose F_{0A}.
 \tag{4.17}\label{eq:global-routes-main}
\]
There is an isotopy \(\mathscr H_s\), fixed on the prescribed complete
marking and collar, with
\[
 \mathscr H_0=F_R^{-1}\compose F_L,\qquad
 \mathscr H_1=\id_W.
 \tag{4.18}\label{eq:rigidity-path-main}
\]
Consequently \(\Gamma_s=F_R\compose\mathscr H_s\) has
\(\Gamma_0=F_L\) and \(\Gamma_1=F_R\).  For the common background marking
\(\mathfrak C_{\mathrm{bg}}\),
\[
 [h_{B\starv}\compose h_{0B}]
 = [h_{AC\starv}\compose h_{AAC}\compose h_{0A}]
 \quad\text{in }\pi_0\Diff^+(Y,\mathfrak C_{\mathrm{bg}}).
 \tag{4.19}\label{eq:mcg-pentagon-main}
\]
This equality is induced by the relative isotopy, not by the equality of
the corresponding integer matrices.

\item \emph{Literal zero-weight Walker lift.}
For the Walker mapping cylinders \(H_e=C([h_e],0)\), forward composition
and the two-sided equality \eqref{eq:lag-main} give
\[
 H_{B\starv}H_{0B}
 =H_{AC\starv}H_{AAC}H_{0A}
 =C([h_{B\starv}\compose h_{0B}],0).
 \tag{4.20}\label{eq:walker-main}
\]
Every constituent weight, all three ordered Maslov seam corrections, and
both route weights are literally zero.
\end{enumerate}
\end{theorem}

The theorem is existential, not canonical: it produces one coordinated
system for every supplied protected sector.  It neither asserts uniqueness
nor compares arbitrary choices.  The precise external boundary is collected
in \cref{sec:scope-boundary}.

\subsection{Construction order}
\label{subsec:witness-order}

For later reference, the reader-facing construction order is
\begin{equation}
\begin{tikzpicture}[
  node distance=6mm and 6mm,
  stagebox/.style={draw,rounded corners=3pt,align=center,inner sep=3.5pt,
    text width=31mm,font=\small},
  arr/.style={-{Latex[length=1.8mm]},thick}
]
  \node[stagebox] (vertex) {vertex data};
  \node[stagebox,right=of vertex] (charts) {compatible endpoint charts};
  \node[stagebox,right=of charts] (cut) {cut maps and quotient descent};
  \node[stagebox,below=of cut] (global) {global extensions};
  \node[stagebox,left=of global] (lag) {homology and a common Lagrangian};
  \node[stagebox,left=of lag] (isotopy) {relative isotopy};
  \node[stagebox,below=of isotopy] (walker) {weight-zero Walker relation};
  \draw[arr] (vertex) -- (charts);
  \draw[arr] (charts) -- (cut);
  \draw[arr] (cut) -- (global);
  \draw[arr] (global) -- (lag);
  \draw[arr] (lag) -- (isotopy);
  \draw[arr] (isotopy) -- (walker);
\end{tikzpicture}
\tag{4.21}\label{eq:witness-order-main}
\end{equation}
The endpoint-extension equation is therefore asserted only after the
representative has been selected.  Likewise, the edge-dependent collar
widths restrict \(u_{\mathrm{cmp}}\); they do not redefine \(u_0\), any
\(z_v\), or the domain \(D^2\times[0,u_0]\).