\section{Compatible endpoint extensions and the exceptional-ribbon reserve}
\label{sec:naturalization}

Independent fillings of the two endpoint disks need not agree in the local
model.  The endpoint transition incorporates both fillings, and its relative
extension will transport the complete marking without requiring their gauges
to coincide.  The exceptional-ribbon reserve must also be fixed before that
transition is used to select an edge representative.

The vertex construction of \cref{sec:vertex-geometry} is complete before
the present section begins.  In particular, the five cut balls \(K_u\), the
maps \(q_u^P\), the vertex collars
\[
 z_u:D^2\times[0,u_0]\longrightarrow K_u,
\]
and the complete markings are already fixed.  For expository continuity,
the first subsection spells out the pre-edge exceptional-reserve choice
announced before \cref{lem:three-common-footprint-disks}.  Its logical inputs
are only these vertex data and the named native carrier; it precedes the
choice of \(A_{jk}\) and all endpoint chart-and-marking systems.  From the second subsection
onward, the common local models and ten incidence labels constructed in
\cref{sec:endpoint-atlases} are fixed, and nothing below alters them.

The reserve keeps the longitude correction inside the common local model.
The independent endpoint charts then determine the boundary transition and
its relative extension recipe.  Selection of the five representatives waits
until the ten spectator atlases of \cref{sec:relative-extensions} are fixed,
so both halves of every cut ball already have compatible interface germs.

\subsection{The exceptional \texorpdfstring{\(AC\star\)}{AC*} ribbon}

For an edge \(e:v\to w\), retain the named native carrier of
\cref{sec:endpoint-atlases} under the notation
\[
 \mathcal S_{e,u}^{\mathrm{LS,geom}}
 :=\mathcal S_{e,u}^{\mathrm{nat}}.
\]
It consists of the mover meridian tube, the slide band, the carrier
longitude and its push-off, the listed smoothing boxes, and the
direct-dual ribbons, with their prescribed product germs.  This notation
does not mean the support of an arbitrary interior smoothing of a
cut-ball representative.  At the two \(AC\star\) incidences it also
contains the complete source reserve or terminal raw-to-target sweep.

The terminal longitude is not defined by an edge map.  Before any edge
representative is chosen, the direct dual construction fixes
\[
 L_{ji}=\ell_j\natural_{\alpha^\vee}(-\ell_i)^+,\qquad
 K_{ki}=\ell_k\natural_{\beta_{ki}^\vee}(-\ell_i)^+,\qquad
 K_{kj}=\ell_k\natural_{\beta_{kj}^\vee}(-\ell_j)^+ .
\]
The raw \(AC\star\) update is
\begin{equation}
 \widetilde K
 =K_{ki}\natural_{\beta_{AC\star}^\vee}(-L_{ji})^+ .
 \label{eq:raw-sharp-longitude}
\end{equation}

The exceptional reserve is built inside the direct-dual track and then
enlarged, without rechoice, to the common disk \(A_{jk}\) in
\cref{lem:three-common-footprint-disks}.  Let
\(
 V_5=\{0,B,A,AC,\starv\}
\), and let \(\mathcal U_{\mathrm{DQ}}^{\mathrm{top}}\) be the frozen open
direct-dual longitude reserve containing the cancelling
\(i^{-1}i\)-track.  Before any edge, choose closed plumbing boxes
\begin{equation}
 R^{\mathrm{top}}\Subset\operatorname{int}R^{+,\mathrm{top}},
 \qquad
 \overline{R^{+,\mathrm{top}}}
 \Subset\mathcal U_{\mathrm{DQ}}^{\mathrm{top}}.
 \label{eq:ear-buffered-top-boxes}
\end{equation}
The larger box still misses every meridian disk and meridian-band germ.
Cutting therefore carries the boxes, without splitting them, to typed
patches
\begin{equation}
 R_v\Subset\operatorname{int}R_v^+\subset\partial K_v,
 \qquad v\in V_5.
 \label{eq:ear-buffered-row-boxes}
\end{equation}
Thus every avoidance assertion below is row-typed inside a single
\(\partial K_v\).  At the terminal row \(R_{\starv}\) is the rectangle
called \(R\) below; \(R_{\starv}^+\) is only its support buffer and does
not enlarge the moving ear or weaken literal equality off \(R\).

Remove the participating subribbons before forming a forbidden set.  The
row-typed participant union \(\mathfrak P_v^\partial\) is
\begin{equation}
\begin{array}{c|p{0.83\linewidth}}
 v&\(\mathfrak P_v^\partial\)\\ \hline
 0,A&\(\varnothing\),\\
 B&the designated \(K_{kj}\)-subribbon in \(R_B\),\\
 AC&the exact \(K_{ki}\)- and \((-L_{ji})^+\)-subribbons and the
  \(\beta_{AC\star}^\vee\) joining strip used by \eqref{eq:raw-sharp-longitude},\\
 \starv&the raw and terminal subribbons and their complete swept
  surface strip.
\end{array}
\label{tab:exceptional-ear-participants}
\end{equation}
These are alternatives in different cut spheres.  In particular, the
participating pieces of \(K_{ki}\) and \(L_{ji}\) at \(AC\) are allowed
to enter \(R_{AC}\); their common portions outside that box remain fixed.

For each row let \(\mathfrak F_v^\partial\) be the union of the following
closed boundary footprints after deleting \(\mathfrak P_v^\partial\):
\begin{enumerate}[label=\textup{(F\arabic*)}]
\item the global \(E\)-footprint; the complete disk footprints of
\[
 i_0,j_0,k_0,j_\varepsilon,k_\varepsilon,k_{2\varepsilon},
 J,J^+,I_A,I_\star;
\]
the three meridian bands \(\alpha,\beta_0,\beta_1\); every smoothing box;
and all six signed face-collar footprints \(D_{r,v}^{\pm}\),
\(r\in\{i,j,k\}\);
\item every complete cut-dual ribbon not in
\(\mathfrak P_v^\partial\), so at \(\starv\) this includes the fixed
\(\ell_i\)- and \(L_{ji}\)-ribbons, while at \(AC\) it excludes precisely
the participating pieces in the raw word;
\item all fifteen auxiliary spoke ribbons, their attachment sectors on
global \(E\) and the signed faces, and their reserved ribbon tubes;
\item all five detector germs with both framing vectors, every two-sided
signed-face pairing footprint, and every protected complementary-disk
annulus;
\item the global \(E\)-bicollar footprint and every nonparticipating
interface-side reserve among
\[
\begin{gathered}
 E_{0B,0},E_{0B,B},E_{B\starv,B},E_{B\starv,\starv},
 E_{0A,0},E_{0A,A},\\
 E_{AAC,A},E_{AAC,AC},E_{AC\starv,AC},E_{AC\starv,\starv}.
\end{gathered}
\]
At this stage these are only boundary or side-reserve roles: no future
local-base interior, depth cap, or endpoint witness is being imported.
\end{enumerate}

For each nonempty participant row choose a closed rounded reserve
\begin{equation}
 \mathcal S_{\mathrm{res},v}^\partial
 \Subset\operatorname{int}R_v^+
 \label{eq:ear-row-surface-reserves}
\end{equation}
containing exactly the participating source strips at \(AC\), the
designated strip at \(B\), and at \(\starv\) both endpoint strips, all
intermediate strips, and the common tails.  The common tails may meet
\(\partial R_v\), but the whole reserve lies compactly in \(R_v^+\).
The routing is chosen jointly inside the open direct-dual reserve so these
sets miss \(\mathfrak F_v^\partial\).  Fixed boundary metrics and the
finite number of rows give the typed clearance
\begin{equation}
 \delta_\partial
 =\min_{\mathcal S_{\mathrm{res},v}^\partial\ne\varnothing}
 \operatorname{dist}_{\partial K_v}
 \bigl(\mathcal S_{\mathrm{res},v}^\partial,
       \mathfrak F_v^\partial\bigr)>0.
 \label{eq:ear-boundary-clearance}
\end{equation}
Both arguments of each distance lie in the same boundary sphere.

Use the retained depth \(d_{\mathrm{prod}}\) from
\eqref{eq:exceptional-depth-chain}, which lies below every relevant
positive product-germ depth.  In the terminal cut ball set
\begin{equation}
 \widehat{\mathcal S}_{\mathrm{res}}
 =c_{\starv}\bigl(
   \mathcal S_{\mathrm{res},\starv}^\partial
   \times[0,d_{\mathrm{prod}}]\bigr),
 \label{eq:ear-lifted-terminal-reserve}
\end{equation}
and let \(\widehat{\mathfrak F}_{\starv}^+\) be the finite union of the
closed depth-\(d_{\mathrm{prod}}\) positive inward halves of all
nonparticipating terminal germs in (F1)--(F5).  Product separation and
\eqref{eq:ear-boundary-clearance} give
\begin{equation}
 \delta_{\mathrm{amb}}
 =\operatorname{dist}_{K_{\starv}}
 \bigl(\widehat{\mathcal S}_{\mathrm{res}},
       \widehat{\mathfrak F}_{\starv}^+\bigr)>0.
 \label{eq:ear-ambient-clearance}
\end{equation}
The negative halves of the two-sided pairings stay in their local doubles;
their boundary traces occur in \(\mathfrak F_{\starv}^\partial\), while
their realized positive halves occur in
\(\widehat{\mathfrak F}_{\starv}^+\).  Thus no negative germ is placed
illegally inside \(K_{\starv}\).

Now make the simultaneous final pre-edge scale choice announced before
\cref{lem:three-common-footprint-disks}.  First choose the coordinate width
\(a\) and the moving surface-ribbon radii inside the open reserve so that
the ensuing compact clearances remain positive.  Then choose the final
restrictions of \(h_0,n_0,n_1,c_P\), together with the subordinate
\(\epsilon_{\mathrm{work}},u_0,\delta_{\mathrm{rig}}\), the restricted
source pairing depth \(\eps_{\mathrm{pair}}^{\mathrm{src}}\), and
\(\epsilon_c\), so that
\begin{equation}
 \begin{gathered}
  3c_P<n_0<n_1<h_0,\qquad u_0=c_P/2,\qquad
  0<\epsilon_c<\eps_{\mathrm{pair}}^{\mathrm{src}}
  <\delta_{\mathrm{rig}}<\epsilon_{\mathrm{work}}<c_P,\\
  20h_0<\min\{\delta_\partial,\delta_{\mathrm{amb}},d_{\mathrm{prod}}\},\\
  \text{every moving surface-ribbon radius is less than }
  \delta_\partial/20.
 \end{gathered}
 \label{eq:ear-clearance-scales}
\end{equation}
The open reserve permits \(a\) to be chosen sufficiently small that these
nested inequalities are compatible with \(c_P\ll a\ll h_0\).  This is one
initial finite choice, made before the common disks and endpoint chart-and-marking systems;
it is not a later change to the already fixed vertex data.  It preserves
\eqref{eq:vertex-depth-hierarchy} and
\eqref{eq:exceptional-depth-chain}.

For completeness, let \(N_{\mathrm{sweep}}^\partial\) denote the compact
terminal surface strip swept by the explicit family \(\rho_t\) below,
including its common tails.  The simultaneous direct-dual choice puts
\[
 b_{AC}^{-1}(\mathcal S_{\mathrm{res},AC}^\partial)
 \quad\text{and}\quad
 b_{\starv}^{-1}(N_{\mathrm{sweep}}^\partial)
\]
in the same protected \(j/k\)-arm of \(\mathsf W\).  Choose the truncated
\(j\)- and \(k\)-branch trees, their spokes stopped before global \(E\),
and a protected connector through both compact sets while avoiding the
\(i\)-branch.  The choices are made jointly so that, after retracting the
ribbon reserves to their core tracks, the resulting spine is a tree and
the unretracted union has a disk regular neighborhood.  A slightly enlarged
closed rounded such neighborhood is a disk
\begin{equation}
 A_{jk}^{\mathrm{ear}}\Subset\mathsf W,
 \qquad
 b_{AC}^{-1}(\mathcal S_{\mathrm{res},AC}^\partial)
 \cup b_{\starv}^{-1}(N_{\mathrm{sweep}}^\partial)
 \Subset\operatorname{int}A_{jk}^{\mathrm{ear}}.
 \label{eq:ear-model-disk-construction}
\end{equation}
This construction uses only the common pre-edge surface and the already
fixed maps \(b_{AC},b_{\starv}\).  Put
\begin{equation}
\begin{aligned}
 A_{jk,v}^{\mathrm{ear}}&=b_v(A_{jk}^{\mathrm{ear}}),\\
 \widehat{\mathcal S}_{AC}^{\mathrm{ear}}
 &=c_{AC}\bigl(
   \mathcal S_{\mathrm{res},AC}^\partial\times[0,n_1]\bigr),\\
 E_{AC\starv,v}^{\mathrm{ear}}
 &=c_v(A_{jk,v}^{\mathrm{ear}}\times\{2h\})
   \cup c_v(\partial A_{jk,v}^{\mathrm{ear}}\times[0,2h]),
 \qquad v\in\{AC,\starv\}.
\end{aligned}
\label{eq:ear-physical-side-and-cap}
\end{equation}
Include \(\partial A_{jk,v}^{\mathrm{ear}}\) in the corresponding
interface-side forbidden footprint in (F5), and include
\[
 c_{\starv}(\partial A_{jk,\starv}^{\mathrm{ear}}
            \times[0,d_{\mathrm{prod}}])
 \subset\widehat{\mathfrak F}_{\starv}^+.
\]
Take the minima in \eqref{eq:ear-boundary-clearance} and
\eqref{eq:ear-ambient-clearance} after these additions.  The compact margin
in \eqref{eq:ear-model-disk-construction} keeps both endpoint reserves away
from these sides, and \(n_1<h<d_{\mathrm{prod}}<2h\) keeps the moving
support away from the caps.  After this pre-edge disk is fixed,
\cref{lem:three-common-footprint-disks,lem:exceptional-ear-clearance}
enlarge it to the actual endpoint base and bind it to the endpoint-specific local
ball by
\[
 A_{jk}^{\mathrm{ear}}\Subset\operatorname{int}A_{jk},
 \qquad b_v^\partial|_{A_{jk}^{\mathrm{ear}}}
 =b_v|_{A_{jk}^{\mathrm{ear}}}
\]
and the already fixed collar \(c_v\) supplies the common product coordinate
through depth \(2h\).  Thus the surface reserve and its inward sweep lie in
the same endpoint-specific local ball.

\begin{lemma}[Exceptional-ribbon reserve and isotopy]
\label{lem:exceptional-ear}
The fixed direct-dual choices come with the closed rounded disk
\(A_{jk}^{\mathrm{ear}}\) of
\cref{lem:three-common-footprint-disks} and admit an ambient isotopy
\(\widehat H_t\) such that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item
 \(A_{jk}^{\mathrm{ear}}\Subset\Int A_{jk}\), and the source and terminal
 ribbon sweeps have compact base projection in
 \(A_{jk}^{\mathrm{ear}}\);
\item
 \(\widetilde K\) and \(K_{kj}\), including their parameterized ribbon and
 inward-normal germs, agree literally outside one closed rectangle \(R\);
\item
 on the whole protected ribbon germ,
 \begin{equation}
  \widehat H_t\widehat\rho_0(r,s,n)
  =\widehat\rho_t(r,s,n),\qquad
  |r|\le r_1,\quad 0\le n\le n_0 ,
  \label{eq:sharp-full-germ}
 \end{equation}
 so the longitudinal parameter \(s\), surface-normal parameter \(r\), and
 inward-normal parameter \(n\) retain their meanings throughout;
\item
 \(\widehat H_1\widehat\rho_0=\widehat\rho_{kj}\), its support is contained
 at depth \(<n_1<h_0<h\), and it misses the side and cap of the actual
 \(AC\star\) local ball as well as every nonparticipating protected germ.
\end{enumerate}
The target \(K_{kj}\) is fixed before \(\widehat H_t\); the arrow is from
the raw ribbon to that pre-existing target ribbon, never conversely.
\end{lemma}

\paragraph{Bigon interpolation in the reserved disk.}
The raw and terminal ribbons have common tails, so the discrepancy is
localized to one empty bigon in the reserved disk.  An endpoint-flat
interpolation of that bigon lifts through the fixed collar to an ambient
flow; the depth cutoff preserves the full ribbon germ, while the clearance
inequalities keep the motion away from the side, cap, and nonparticipating
data.  The final endpoint binding therefore changes only the exceptional
raw-ribbon component.

\begin{proof}
\emph{A boundary ribbon isotopy.}
The oriented traversal of \eqref{eq:raw-sharp-longitude} is
\[
 \ell_k\longrightarrow(-\ell_i)^+
 \longrightarrow(\ell_i)^+\longrightarrow(-\ell_j)^+ .
\]
The two middle tracks are adjacent and oppositely oriented.  Route the raw
and terminal alternatives on the same \(\ell_k\)-tail and
\((-\ell_j)^+\)-tail, and place the intervening empty bigon in a rectangle
\[
 R=[-2,2]_x\times[7a/2,11a/2]_u.
\]
With the endpoint-flat function
\[
 \psi(s)=
 \begin{cases}
 \exp\!\left(1-\dfrac1{1-s^2}\right),& |s|<1,\\
 0,& |s|\ge1,
 \end{cases}
\]
the middle arcs
\[
 \gamma_t(s)=
 \left(-1+2(1-t)\psi(s),\,\frac{9a}{2}+\frac a2s\right),
 \qquad -1\le s\le1,
\]
are embedded, agree to every order at their endpoints, and interpolate
from the reserve arc to the straight terminal arc.  Thickening them by
\[
 \rho_t(r,s)=
 \left(-1+2(1-t)\psi(s)+r,\,\frac{9a}{2}+\frac a2s\right)
\]
gives orientation-preserving ribbon charts because
\[
 \det\frac{\partial(x,u)}{\partial(r,s)}=\frac a2>0.
\]
More precisely, if \(T_{\mathrm{tail}}^\partial\) is the union of the two
common closed tail strips, then
\begin{equation}
 N_{\mathrm{sweep}}^\partial
 =T_{\mathrm{tail}}^\partial
  \cup\bigcup_{0\le t\le1}
   \rho_t([-r_1,r_1]\times[-1,1])
 \Subset\mathcal S_{\mathrm{res},\starv}^\partial.
 \label{eq:exact-ear-surface-sweep}
\end{equation}
It is compact, and \eqref{eq:ear-boundary-clearance} separates it from
every nonparticipating disk, ribbon, spoke, detector, pairing, annular,
and global-interface footprint.  Its positive product lift lies in
\(\widehat{\mathcal S}_{\mathrm{res}}\), so
\eqref{eq:ear-ambient-clearance} gives the corresponding ambient
separation.

\emph{Extension through the fixed collar.}
On the moving ribbon tube use the time-dependent field
\(
 V_t(\rho_t(r,s))=-2\psi(s)\partial_x
\),
cut off before the boundary of the reserve.  Its flow \(H_t\) satisfies
\(H_t\rho_0=\rho_t\) by uniqueness of integral curves.  Lift it through
the fixed endpoint collar \(c_\star\), with a depth cutoff equal to one on
\([0,n_0]\) and zero from \(n_1\) onward.  The lifted field has no
\(\partial_n\)-component, so its flow gives
\eqref{eq:sharp-full-germ} and preserves both framing vectors.  Flatness
of \(\psi\) gives literal agreement of all tangential and normal derivatives
with the fixed tails.

\emph{Comparison with the fixed endpoint map.}
Finally, \eqref{eq:ear-model-disk-construction} puts both endpoint sweep
preimages compactly in the interior of the pre-edge reserve disk.  Its
boundary was included in the forbidden set before the final clearance
minima were taken.  The endpoint binding displayed above agrees with the
fixed vertex maps on that disk and uses their common collar through depth
\(2h\).  The strict inequalities
\[
 3c_P<n_0<n_1<h_0<h<2h
\]
together with \eqref{eq:ear-clearance-scales} therefore place the closed
source and terminal sweeps away from the actual side and cap and from all
nonparticipating protected germs.  This is the required binding to
\(A_{jk}\), rather than an assumption imported from the preliminary
sharp-row construction.
\end{proof}

\subsection{Independent native atlases}

Recall from \eqref{eq:ten-endpoint-incidences} that
\[
 \mathcal I_{10}=\{(e,u):e\in E_5,\ u\text{ is an endpoint of }e\}.
\]
The common local boundary model has the finite face list
\[
 E_e,\ p^-,p^+,\ q^-,q^+;\qquad
 C_1,\ldots,C_{16};\qquad
 R_{s_p},R_{c_p},R_{s_q},R_{c_q};\qquad Z_e .
\]
Here \(p,q\) are the moving and carrier roles and \(Z_e\) is the unique
unmarked complementary disk.

\begin{proposition}[Independent endpoint chart-and-marking systems]
\label{prop:native-atlases}
For every \((e,u)\in\mathcal I_{10}\), the atlas already fixed in
\eqref{eq:ten-native-boundary-atlases} is an orientation-preserving full
boundary atlas
\begin{equation}
 \widehat h_{e,u}:
 \partial K_{e,\mathrm{mod}}^{\mathrm{loc}}
 \longrightarrow \partial B_{e,u}^{\mathrm{LS}} .
 \label{eq:native-endpoint-atlas}
\end{equation}
The ten atlases were fixed independently of all edge representatives and
of the atlas at the other endpoint.  They intertwine the two abstract
pairings with the endpoint pairings, use the common oriented interface
coordinate, and contain a full positive disk chart
\[
 \widehat\zeta_{e,u}:D^2\longrightarrow Z_{e,u}^{\mathrm{LS}}
\]
whose germ is fixed on the whole overlap annulus.  Their residual freedom
is exactly
\begin{equation}
 \prod_{(e,u)\in\mathcal I_{10}}
 \Diff^+(D^2\mathbin{\mathrm{rel}}\mathcal A_{e,\mathrm{mod}}).
 \label{eq:native-residual-gauge}
\end{equation}
\end{proposition}

\begin{proof}
Its construction maps the first five faces, sixteen collar rectangles,
and four ribbon rectangles through the corresponding endpoint's own fixed product
charts.  On overlaps, the two incident faces use the same tangent
parameter and opposite boundary-side designation.  The signed pairings
are tied to the same abstract involutions, and the interface charts use
one oriented disk coordinate.  Thus these maps form a smooth protected
atlas rather than a list of unrelated face maps.

Concatenate the sixteen oriented edge parameters to
\(\theta\in\R/\Z\), assigning the \(m\)-th edge the interval
\([ (m-1)/16,m/16]\).  Together with the common radial variable this
gives the full annular chart.  Fill the remaining disk independently at
each endpoint, relative to an open neighborhood of that annulus.  All
other charts were already fixed, so the transition between two possible
fillings is precisely a factor in \eqref{eq:native-residual-gauge}.
\end{proof}

For the two \(AC\star\) incidences, the source atlas contains the direct
\(L_{ji}\)- and \(K_{ki}\)-rows and hence the raw ribbon
\(\widetilde K\).  The target atlas contains the independently fixed
\(K_{kj}\)-row.  The map \(\widehat H_1\) from
\cref{lem:exceptional-ear} is used only when the target endpoint is
identified with its endpoint-specific local ball.

\subsection{The common template and quotient-compatible endpoint extensions}

Using the rounded endpoint chart already fixed at endpoint \(u\), retain
\[
 G_{e,u}^{\mathrm{loc}}(x,t)=c_u(b_u(x),t),\qquad
 g_{e,u}^{\mathrm{loc}}
 =G_{e,u}^{\mathrm{loc}}\big|_{\partial K_{e,\mathrm{mod}}^{\mathrm{loc}}}.
\]
The pair already fixed, before either endpoint extension and before any
representative, in \eqref{eq:common-endpoint-compression} is
\begin{equation}
 \ell_e>0,\qquad
 \lambda_e:[0,h_0)\longrightarrow[0,\ell_e),\qquad
 \lambda_e(0)=0,\quad \lambda_e'>0,\quad
 \lambda_e(t)=t\ \text{near }0,
 \label{eq:nat-template}
\end{equation}
with
\(\lambda_e([0,\epsilon_{\mathrm{work}}])\Subset[0,\ell_e)\).
The same pair
\((\ell_e,\lambda_e)\) is used at both endpoints.

\begin{proposition}[Endpoint extension and native shallow identity]
\label{prop:nat-template-qhat}
The endpoint-only boundary map of
\eqref{eq:endpoint-only-boundary-identification} is
\[
 \widehat a_{e,u}^{\mathrm{loc}}
 =g_{e,u}^{\mathrm{loc}}(\widehat h_{e,u})^{-1}.
\]
The orientation-preserving diffeomorphism fixed in
\cref{lem:relative-local-ball-extension},
\[
 \widehat\Theta_{e,u}:
 B_{e,u}^{\mathrm{LS}}\longrightarrow K_{e,u}^{\mathrm{loc}},
\]
has the full collar formula
\begin{equation}
 \widehat\Theta_{e,u}
 \bigl(\widehat h_{e,u}(x),\lambda_e(t)\bigr)
 =G_{e,u}^{\mathrm{loc}}(x,t)
 \label{eq:theta-collar}
\end{equation}
and is coordinate identity on the signed local interface, with all jets.
After \(\widehat\Theta_{e,u}\) is fixed, the shallow map of
\eqref{eq:native-shallow-embedding} is
\[
 \widehat q_{e,u}^{\mathrm{LS}}(x,t)
 =\bigl(\widehat h_{e,u}(x),\lambda_e(t)\bigr)
\]
on the common local shallow domain.  Then
\begin{equation}
 \widehat\Theta_{e,u}^{-1}q_u^P
 =\widehat q_{e,u}^{\mathrm{LS}} .
 \label{eq:qhat}
\end{equation}
Moreover, the transported named carrier satisfies
\begin{equation}
 \mathcal S_{e,u}^{\mathrm{LS,phys}}
 :=\widehat\Theta_{e,u}
   (\mathcal S_{e,u}^{\mathrm{LS,geom}})
 \subset c_u(A_{e,u}\times[0,h))
 \Subset K_{e,u}^{\mathrm{loc}} .
 \label{eq:named-carrier-transport}
\end{equation}
\end{proposition}

\begin{proof}
In the two full atlases,
\(\widehat a_{e,u}^{\mathrm{loc}}\) is coordinate identity on the
protected region, the full complementary disk, their whole annular
overlap, and the signed interface.  At the exceptional target, compose
only the raw-ribbon component with \(\widehat H_1\); by
\cref{lem:exceptional-ear} this changes no other endpoint datum.  The
hypotheses of the already established relative local-ball extension
\cref{lem:relative-local-ball-extension} are therefore satisfied
endpoint by endpoint.  Its conclusion is exactly
\eqref{eq:theta-collar}.

The definition of \(\widehat q_{e,u}^{\mathrm{LS}}\) is made only after
that extension exists.  On the common local shallow half,
\(G_{e,u}^{\mathrm{loc}}=q_u^P\), and
\eqref{eq:qhat} follows by applying
\(\widehat\Theta_{e,u}^{-1}\) to \eqref{eq:theta-collar}.  The identity
thus uses only the data at this endpoint, independently of the opposite
endpoint and any edge representative.

The named carrier has pre-edge base closure in \(\Int A_{e,u}\) and
native depth \(<h_0<h\).  Formula \eqref{eq:theta-collar} therefore gives
\eqref{eq:named-carrier-transport}.  For the exceptional incidences the
side-and-cap control comes from \cref{lem:exceptional-ear}.  No statement
about the support of the later interior smoothing is made.
\end{proof}

\subsection{The uninstantiated extension schema}

Only after the two endpoint constructions in
\cref{prop:nat-template-qhat} exist is the boundary transition already
fixed in \eqref{eq:post-atlas-boundary-transition} retained as
\begin{equation}
 \widehat\phi_e^\partial
 :=\widehat h_{e,w}(\widehat h_{e,v})^{-1}:
 \partial B_{e,v}^{\mathrm{LS}}
 \longrightarrow\partial B_{e,w}^{\mathrm{LS}} .
 \label{eq:nat-boundary-transition}
\end{equation}
It sends every protected row by the common label dictionary and absorbs
the two independent complementary-disk gauges.  Indeed, if
\(\widehat\zeta_{e,u}=\zeta_{e,u}^0\psi_{e,u}\), then its disk row is
\[
 \zeta_{e,w}^0\psi_{e,w}\psi_{e,v}^{-1}
   (\zeta_{e,v}^0)^{-1}.
\]
Thus neither gauge is synchronized with the other.

\begin{proposition}[Compatible endpoint-extension schema]
\label{prop:nat-schema}
The boundary map \eqref{eq:nat-boundary-transition} is
orientation preserving, intertwines separately the two source active
pairings with their corresponding target pairings,
is the identity in the signed interface coordinate, and lies in the
boundary relative class of the positive two-site slide from
\cref{thm:positive-two-site-input}.  It determines a recipe for choosing
an orientation-preserving cut-ball representative in that theorem's
\(B0/B5\) class, with the same full collar formula and all marked product
germs.
\end{proposition}

\paragraph{Relative Alexander isotopy on the remaining disk.}
The two endpoint chart-and-marking systems already agree on the protected cellulation and
annulus, so their quotient is confined to the remaining disk.  A relative
Alexander isotopy identifies that disk map with the positive local boundary
class; product extension across a collar and relative approximation then
produce the smooth cut-ball representative while preserving every prescribed
germ.

\begin{proof}
Smoothness, orientation, pairing compatibility, and the interface formula
follow directly from the two independently constructed atlases.  On the
protected cellulation the map is the cellular B0 dictionary.  On the
complementary disk, its quotient with the B0 boundary reference fixes the
whole overlap annulus.  The explicit radial Alexander isotopy on the inner
disk therefore gives a topological isotopy relative to the protected
region; no smooth relative isotopy is asserted.

Extend the boundary map as a product on a full collar and radially extend
the remaining inner sphere map.  Denote the resulting ball homeomorphism
by \(\widehat\Phi_e^{\mathrm A}\), and double it across the boundary.  The
doubled map is already smooth on a full
two-sided equatorial neighborhood.  Let \(K_e\) be the equator together
with a closed thinner two-sided collar containing all prescribed product
germs.  Applied directly to this doubled homeomorphism, the classical relative
homeomorphism-approximation theorem, in the form stated by Pardon
\cite[Theorem~4.6]{Pardon2020}, gives a
diffeomorphism of the source double to the target double which equals the
doubled map on a neighborhood of \(K_e\).

The equator separates both doubles.  Since the new diffeomorphism is the
side-preserving product map near the equator, connectedness sends the
chosen source side into the chosen target side; applying the same argument
to the inverse proves equality of those sides.  Restriction gives a
temporary smoothed recipe output \(\widehat\Phi_e^{\mathrm{sm}}\) with
all prescribed product germs.  Its collar determinant is positive, hence
so is its determinant everywhere by connectedness.

Here is the full three-dimensional B0/B5 comparison.  The maps
\(\widehat\Phi_e^{\mathrm A}\) and
\(\widehat\Phi_e^{\mathrm{sm}}\) agree on the same full boundary collar.
After deleting a thinner open collar, identify the remaining source with
the standard closed ball and form
\[
 Q_e=(\widehat\Phi_e^{\mathrm{sm}})^{-1}
       \widehat\Phi_e^{\mathrm A}.
\]
Its restriction to the inner ball fixes the boundary pointwise.  For
\(0<s\le1\), the dimension-three Alexander formula
\[
 \mathcal H_{e,s}(x)=
 \begin{cases}
  sQ_e(x/s),&|x|\le s,\\
  x,&s\le |x|\le1,
 \end{cases}
 \qquad \mathcal H_{e,0}(x)=x
\]
is a jointly continuous isotopy from the identity to \(Q_e\), with inverse
given by the same formula using \(Q_e^{-1}\).  Extend it by the identity
over the deleted collar.  Then
\(\widehat\Phi_e^{\mathrm{sm}}\mathcal H_{e,s}\) is a topological isotopy,
relative to the full prescribed collar, from the smoothed recipe output to
\(\widehat\Phi_e^{\mathrm A}\).

The two-dimensional Alexander path constructed above compares the boundary
of \(\widehat\Phi_e^{\mathrm A}\) with the B0 cellular boundary map while
fixing the protected region, the complete overlap annulus, both pairing
equations, and the signed interface germ.  At each parameter extend that
boundary path as a product on the same collar and use the radial Alexander
extension on the complementary inner ball.  The collar and radial formulas
agree at their seam and vary continuously, giving a topological ball
isotopy, relative to all those protected data, from
\(\widehat\Phi_e^{\mathrm A}\) to the B0 ball extension.  Concatenating the
two ball isotopies proves that the smoothed recipe output lies in exactly
the specified B0/B5 relative class.  Only a topological relative isotopy is
used; no smooth relative-isotopy uniqueness is asserted.
\end{proof}

\begin{warning}[No representative has yet been selected]
\label{warn:nat-schema-no-representative}
At this stage \eqref{eq:nat-boundary-transition} and
\cref{prop:nat-schema} specify a deterministic construction recipe only.
The symbol \(\widehat\phi_e^{\mathrm{cut}}\) is not bound until all ten
spectator atlases have been constructed in
\cref{prop:spectator-atlases}.  Consequently no shallow, pairing, or
interface equation involving that representative is used in this
section.
\end{warning}