\section{Quotient descent and global extension}
\label{sec:descent-global}

The full-normal pairing identities make the cut-ball map smooth after the
three disks are reglued.  Its identity formula near the exterior attachment
then permits extension to the compression body by the identity on the old
region and unused handles.  These two operations produce the quotient and
global maps from the already constructed cut map.

Throughout, \(e:v\to w\) is fixed and the pairing maps are those of
\cref{prop:pair15}.  We first descend through the central faces, then identify
the quotient map on the inside half of the global interface, and finally
adjoin the exterior identity.  The terminal stabilization statement retains
the distinct domains of the three maps and their marking conditions.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[
  obj/.style={draw,rounded corners=3pt,inner sep=4pt},
  arr/.style={-{Latex[length=2mm]},thick},
  every node/.style={font=\small}
]
  \node[obj] (Kv) at (0,1.2) {\(K_v\)};
  \node[obj] (Kw) at (4.6,1.2) {\(K_w\)};
  \node[obj] (Uv) at (0,0) {\(U_{ijk,v}\)};
  \node[obj] (Uw) at (4.6,0) {\(U_{ijk,w}\)};
  \node[obj] (Wv) at (0,-1.2) {\((W,\mathfrak C_v)\)};
  \node[obj] (Ww) at (4.6,-1.2) {\((W,\mathfrak C_w)\)};
  \draw[arr] (Kv) -- node[above] {\(\widetilde F_e\)} (Kw);
  \draw[arr] (Uv) -- node[above] {\(f_{e,U}\)} (Uw);
  \draw[arr] (Wv) -- node[above] {\(F_e\)} (Ww);
  \draw[arr] (Kv) -- node[left] {\(\rho_v^{(3)}\)} (Uv);
  \draw[arr] (Kw) -- node[right] {\(\rho_w^{(3)}\)} (Uw);
  \draw[arr] (Uv) -- node[left=24pt,align=right] {extend by identity\\across \(E\)} (Wv);
  \draw[arr] (Uw) -- node[right=24pt,align=left] {extend by identity\\across \(E\)} (Ww);
  \node[align=center] at (2.3,-2.05)
    {full-normal pairing identities make the top square smooth through every seam};
\end{tikzpicture}
\caption{Cut map, quotient descent, and global extension.  The commuting
square is \eqref{eq:quotient-descent}; smoothness uses the two directed
full-normal pairing identities, not only equality on the central faces.
The lower arrows record extension by the identity on the fixed exterior.}
\label{fig:full-normal-collar-descent}
\end{figure}

\subsection{Smooth descent through the three paired faces}

Let \(\sim_a\) be the equivalence relation on \(K_a\) generated by the
three central signed-face identifications
\begin{equation}
 \kappa_{r,a}^{-,(3)}(z,0)
 \sim_a
 \Pi_{r,a}^{-+,(3)}\kappa_{r,a}^{-,(3)}(z,0),
 \qquad r\in\{i,j,k\}.
 \label{eq:central-face-equivalence}
\end{equation}
Only the central faces are identified in the quotient.  The positive-depth
extensions of the pairing maps are seam charts used to prove smoothness;
they are not additional quotient generators.  Write
\begin{equation}
 \rho_a^{(3)}:K_a\longrightarrow
 U_{ijk,a}=K_a/\!\sim_a
 \label{eq:quotient-map-section10}
\end{equation}
for the quotient.  Its exposed-boundary trace is \(\rho_a\).

\begin{proposition}[Quotient descent]
\label{prop:quotient-descent}
There is a unique orientation-preserving diffeomorphism
\(\Fquot e:U_{ijk,v}\to U_{ijk,w}\) satisfying
\begin{equation}
 \Fquot e\rho_v^{(3)}=\rho_w^{(3)}\Fcut e.
 \label{eq:quotient-descent}
\end{equation}
Its inverse is the descent of \((\Fcut e)^{-1}\), and on the exposed
boundary
\begin{equation}
 (\Fquot e|_{\partial_+U_{ijk,v}})\rho_v
 =\rho_w(\Fcut e|_{\partial K_v}).
 \label{eq:quotient-boundary-trace}
\end{equation}
\end{proposition}

\begin{proof}
The pairing conjugacies have two roles: they make the quotient formula
independent of the chosen representative and make it smooth across each
reglued disk.

Define
\[
 \Fquot e([x]_v)=[\Fcut e(x)]_w.
\]
It is enough to check a generating pair.  If
\(x=\kappa_{r,v}^{-,(3)}(z,0)\) and
\(y=\Pi_{r,v}^{-+,(3)}(x)\), then \eqref{eq:pair-forward} gives
\[
 \Fcut e(y)
 =\Pi_{r,w}^{-+,(3)}\Fcut e(x),
\]
so the two images represent the same \(w\)-class.  The backward identity
\eqref{eq:pair-backward} gives the same conclusion for the opposite
directed generator.  Thus the definition is independent of the chosen
representative.  Surjectivity of \(\rho_v^{(3)}\) gives uniqueness and
\eqref{eq:quotient-descent}.

Away from a reglued disk, the quotient chart is an ordinary cut-ball
chart, so smoothness follows from \cref{prop:cut-map-gluing}.  Near a
reglued disk, use the common full-normal coordinates of
\eqref{eq:source-full-normal-pairing}.  The two representatives have the
same normal variable, and \cref{prop:pair15} says that both expressions
for the descended map, together with every derivative, coincide.  Hence
the quotient formula is smooth through the seam.  Repeating the argument
for \((\Fcut e)^{-1}\) produces a smooth inverse.  Because the cut map is
positive and the quotient normal is preserved, the descended map is
orientation preserving.  Restricting \eqref{eq:quotient-descent} to the
exposed boundary gives \eqref{eq:quotient-boundary-trace}.
\end{proof}

\subsection{The global \texorpdfstring{\(E\)}{E}-seam}

Let \(\cN_a^{\mathrm{in}}(E)\subset U_{ijk,a}\) be the image under
\(\rho_a^{(3)}\) of the inside half of the cut chart
\(\epsilon_{E,e,a}^{\mathrm{cut}}\).  The supplied exterior marking
identifies these endpoint neighborhoods, in their complete product
coordinates, with one fixed inside-half neighborhood
\(\cN^{\mathrm{in}}(E)\).  Descending \eqref{eq:global-E-cut} yields
\begin{equation}
 \Fquot e|_{\cN^{\mathrm{in}}(E)}=\id.
 \label{eq:global-E-inside-identity}
\end{equation}

Let \(\cN^{\mathrm{out}}(E)\) be the supplied exterior product half and set
\begin{equation}
 \cN^{\mathrm{two}}(E)
 =\cN^{\mathrm{in}}(E)\mathop{\cup}_E\cN^{\mathrm{out}}(E).
 \label{eq:two-sided-global-E-neighborhood}
\end{equation}
Also let \(W_{\mathrm{old}}\) be the fixed old region.  For each
\(t\notin\{i,j,k\}\), let \(\cN_t\) be the fixed product neighborhood of
the \(t\)-th unused handle together with its complete marking.

\begin{proposition}[Global extension and ownership]
\label{prop:global-extension}
The formula
\begin{equation}
 \Fglob e(x)=
 \begin{cases}
  \Fquot e(x),&x\in U_{ijk,v},\\
  x,&x\in W\setminus\Int U_{ijk,v}
 \end{cases}
 \label{eq:global-map-piecewise}
\end{equation}
defines an orientation-preserving complete-marked diffeomorphism
\((W,\mathfrak C_v)\to(W,\mathfrak C_w)\).  It satisfies
\begin{align}
 \Fglob e|_{\cN^{\mathrm{two}}(E)}&=\id,
 \label{eq:global-E-two-sided-identity}\\
 \Fglob e|_{W_{\mathrm{old}}}&=\id,
 \qquad
 \Fglob e|_{\bigcup_{t\notin\{i,j,k\}}\cN_t}=\id,
 \label{eq:old-unused-identity}\\
 (\Fglob e|_{\partial_+W})\rho_v
 &=\rho_w(\Fcut e|_{\partial K_v}).
 \label{eq:global-boundary-conjugacy}
\end{align}
Thus the three maps occur in the fixed order
\begin{equation}
 \Fcut e\longmapsto\Fquot e\longmapsto\Fglob e.
 \label{eq:map-registry-section10}
\end{equation}
\end{proposition}

\begin{proof}
The quotient map is already the identity on the inside of the \(E\)-collar;
extending \(\Fquot e\) by the identity on the exterior therefore uses one
coordinate formula on the full bicollar.

In the two-sided product coordinate
\(E\times(-\delta,\delta)\), the first branch of
\eqref{eq:global-map-piecewise} is the identity for the inside sign by
\eqref{eq:global-E-inside-identity}; the second branch is the identity for
the outside sign by definition.  At the central face, both expressions
and all normal derivatives are those of the same coordinate identity.
Consequently they glue to a smooth map on
\(\cN^{\mathrm{two}}(E)\), proving
\eqref{eq:global-E-two-sided-identity}.  The same argument applied to the
piecewise inverse, using the inverse from
\cref{prop:quotient-descent}, gives a smooth two-sided inverse on \(W\).
Positivity follows from positivity on the protected sector and the
identity exterior branch.

The old and unused regions lie in the exterior branch, proving
\eqref{eq:old-unused-identity}.  On the outgoing boundary,
\eqref{eq:global-map-piecewise} and
\eqref{eq:quotient-boundary-trace} give
\eqref{eq:global-boundary-conjugacy}.  All vertex markings were fixed
before the edge maps, and the cut-level marked germs are transported by
\cref{prop:cut-stabilization}; adjoining the fixed exterior marking
therefore equips \(\Fglob e\) with the complete marking.  Finally, each arrow in
\eqref{eq:map-registry-section10} is a construction from the preceding
map, so none of the three stages can be reselected independently.
\end{proof}

\begin{remark}[The case \(N=3\)]
\label{rem:N3-global-extension}
If \(N=3\), the set of indices \(t\notin\{i,j,k\}\) is empty and hence
\(\bigcup_t\cN_t=\varnothing\).  The last identity in
\eqref{eq:old-unused-identity} is then a valid empty-domain statement.
The old region and the two-sided global-\(E\) neighborhood remain present.
No argument above requires an unused handle.
\end{remark}

\subsection{Terminal stabilization}

\begin{corollary}[Stage-correct stabilization]
\label{cor:terminal-stabilization}
For each edge, the cut-level conclusion of
\cref{prop:cut-stabilization}, the descended global-\(E\) identity, and
the exterior identities of \cref{prop:global-extension} form one
complete stabilization package.  Its cut clauses are owned by
\(\Fcut e\), its quotient clause by \(\Fquot e\), and its global/exterior
clauses by \(\Fglob e\).  The endpoint parameterization \(p_v\) is fixed
metadata and is not asserted to be fixed pointwise by \(\Fglob e\).
\end{corollary}

\begin{proof}
All cut-ball words, ribbons, spokes, detector data, pairings, and the
global-\(E\) cut trace were proved before descent in
\cref{prop:cut-stabilization}.  Equation
\eqref{eq:global-E-inside-identity} adds the descended inside half, and
\eqref{eq:global-E-two-sided-identity} adds the exterior half.  Equation
\eqref{eq:old-unused-identity} is the first statement involving the old
or unused regions.  These domains exhaust the marking fields attached to
the protected sector and its exterior completion, proving the combined
claim with the stated ownership.  Since \(p_v\) is used only to identify
the outgoing boundary with the fixed surface \(Y\), no pointwise
\(\Fglob e\)-invariance of \(p_v\) is needed or implied.
\end{proof}