\section{Relative bands, disks, and decorated-ball extension}
\label{sec:relative-extensions}

The spectator map must match the vertex disk collar in an inner region and
the local construction in an outer region.  Agreement on their middle seam
does not determine the transition on the complete protected annulus required
for relative ball extension.  Starting from the local endpoint maps of
\cref{prop:nat-template-qhat}, we choose the actual annular overlap bands,
then compatible full-disk charts and inward collars.  Their full-annulus
agreement supplies the relative extension hypothesis, allowing extension over
the spectator halves while retaining all protected boundary data.

The capital letters used below name the four disk rows, not additional edge
maps.  The short lower and upper rows are
\(J=j_0\mathbin{\#_{\beta_0}}k_\epsilon\) and
\(J^+=j_\epsilon\mathbin{\#_{\beta_1}}k_{2\epsilon}\); the two longer rows
are \(I_A=i_0\mathbin{\#_\alpha}j_\epsilon\) and
\(I_\star=i_0\mathbin{\#_\alpha}J^+\).
Likewise, \(E\) denotes the global pentagon interface, while \(E_e\) is the
disk interface belonging to the single edge \(e\).  The several collar
depths introduced in this section refine the vertex hierarchy already fixed
in \eqref{eq:vertex-depth-hierarchy} and, at the exceptional edge, the chain
in \eqref{eq:exceptional-depth-chain}; they do not define independent
geometric layers.

\subsection{Relative overlap bands}

Retain the positive disk chart
\[
 \zeta_{\mathrm{mod}}:D^2\xrightarrow{\cong}
 Z_{\mathrm{mod}}\subset S
\]
and let \(r:D^2\to[0,1]\) be the polar radius.  On the model surface put
\[
 \bar r(y)=
 \begin{cases}
  r(\zeta_{\mathrm{mod}}^{-1}y),
     &y\in Z_{\mathrm{mod}},\\
  1,&y\notin\Int Z_{\mathrm{mod}}.
 \end{cases}
\]
This is a continuous surface function; no derivative of \(\bar r\) will
be taken.  The radial pre-edge choice of the work surface is recorded,
without changing it, by
\begin{equation}
 \begin{gathered}
 0<r_{\mathrm{work}}<r_{\mathrm{keep}}<r_P<r_A<1,\\
 \mathsf W=\{y\in S:\bar r(y)>r_{\mathrm{work}}\},\qquad
 \mathcal A_{\mathrm{mod}}^{\mathrm{work}}=\{r>r_{\mathrm{work}}\},\qquad
 \mathcal A_{\mathrm{mod}}^{\mathrm{open}}=\{r>r_{\mathrm{keep}}\},\\
 \mathcal A_{\mathrm{mod}}=\{r\ge r_A\},\qquad
 \mathcal A_{\mathrm{mod}}
 \subset\mathcal A_{\mathrm{mod}}^{\mathrm{open}}
 \Subset\mathcal A_{\mathrm{mod}}^{\mathrm{work}},\qquad
 \zeta_{\mathrm{mod}}(\mathcal A_{\mathrm{mod}}^{\mathrm{open}})
 \subset \mathsf W\cap Z_{\mathrm{mod}}.
 \end{gathered}
 \label{eq:protected-radial-data}
\end{equation}
The full protected annulus in \eqref{eq:protected-radial-data} is
different from the artificial inner/outer seam introduced below.

For each edge, let \(K_e^S\) be the nonempty compact base trace of its
rounded local support and spectator spine, excluding the still-free
complementary-disk fills and annular neighborhoods.  The footprints have
already been chosen in the open work surface, so
\[
 \alpha=\min_{e\in E_5}\min_{y\in K_e^S}\bar r(y)
 >r_{\mathrm{work}}.
\]
Choose, before filling any complementary disk,
\begin{equation}
 r_{\mathrm{work}}<r_0<r_-<r_+<r_1<\alpha,
 \qquad r_c=\frac12(r_{\mathrm{work}}+r_0),
 \label{eq:relative-radii}
\end{equation}
then choose
\[
 0<\eps_{01}<\frac12\min\{r_0-r_{\mathrm{work}},r_--r_0,
 r_+-r_-,r_1-r_+,1-r_1\}
\]
and the open annular germ
\[
 \mathcal N_{01}^D
 =\{r_0-\eps_{01}<r<r_1+\eps_{01}\}
\]
whose closure therefore retains a positive margin from every adjacent
radius.  Write
\[
 \mathcal A_{01}^D=\{r_0\le r\le r_1\},\qquad
 C_a^D=\{r\le a\}.
\]
Put
\[
 r_{cc}=\frac12(r_{\mathrm{work}}+r_c),\qquad
 D_c=C_{r_{cc}}^D,\qquad D_c^{++}=C_{r_c}^D.
\]
Both are new witnesses; \(D_c^{++}\) is the larger safe core disk.

\begin{proposition}[Relative band and disk data]
\label{prop:relative-band-data}
For every edge \(e\) there are:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a label \(\sigma_e\in\{+,-\}\) and a positive full-disk chart
 \[
  j_e^c:C_{r_1}^D\xrightarrow{\cong}
  Z_e^{\mathrm{sp},\sigma_e};
 \]
\item a common pre-extension shallow embedding
 \[
  q_e^{\mathrm{sp}}:
  \mathcal D_{e,\mathrm{sp}}^{\mathrm{work}}
  \hookrightarrow L_{e,\mathrm{mod}}^{\mathrm{sp}};
 \]
\item for each endpoint \(u\), a pre-extension product map
 \(\widehat a_{e,u}^{\mathrm{sp}}\) on an open germ containing
 \(\mathcal A_{01}^D\times[0,d_{\mathrm{ann}})\);
\item realized endpoint disk charts with the same label \(\sigma_e\), satisfying
 \[
  b_u\zeta_{\mathrm{mod}}(C_{r_1}^D)
  =Z_{e,u}^{\mathrm{sp},\sigma_e},\qquad
  b_u\zeta_{\mathrm{mod}}(\partial C_{r_1}^D)
  =\partial Z_{e,u}^{\mathrm{sp},\sigma_e}.
 \]
\end{enumerate}
They satisfy
\begin{align}
 P_e^{\mathrm{sp}}\cap Z_e^{\mathrm{sp},\sigma_e}
 &=j_e^c(\mathcal A_{01}^D),                                     
 \label{eq:relative-band-image}\\
 \widehat a_{e,u}^{\mathrm{sp}}
 q_e^{\mathrm{sp}}(\zeta_{\mathrm{mod}}(x),t)
 &=q_u^P(\zeta_{\mathrm{mod}}(x),t)
  =c_u(b_u\zeta_{\mathrm{mod}}(x),t)                             
 \label{eq:pre-dbx-spectator-product}
\end{align}
for \(x\in\mathcal A_{01}^D\) and
\(0\le t<d_{\mathrm{ann}}\), with all three formulas defined on the
surrounding open product germ.  Moreover,
\[
 j_e^c(D_c^{++})\Subset\Int Z_e^{\mathrm{sp},\sigma_e}
 \quad\text{and}\quad
 j_e^c(D_c^{++})\cap P_e^{\mathrm{sp}}=\varnothing ,
\]
and the same safe disk misses the rounded local support.
\end{proposition}

\begin{proof}
The closed disk \(C_{r_1}^D\) misses the already fixed spine and rounded
local support.  Its connected image lies in one of the two labeled
spectator complementary disks; this defines \(\sigma_e\).  Relatively
refine the still-free annular neighborhood so that its inner boundary is
the image of \(r=r_0\), extend that coordinate across both boundary
circles on \(\mathcal N_{01}^D\), and fill positively across the disk.
The compact separation from the support gives the assertion about
\(D_c^{++}\).

The pre-extension shallow model supplies the common spectator half
\(\mathcal D_{e,\mathrm{sp}}^{\mathrm{work}}\), including the outer-work
part, the portions of both annular disk germs that lie in the work surface,
and the signed internal-interface half.  Bind \(q_e^{\mathrm{sp}}\) there
using \(j_e^c\) on the work-surface part of the selected disk.  The complete
disks are chart domains, not an enlargement of this shallow domain.

Compactness gives a common positive \(d_{\mathrm{ann}}\) for which the
whole annulus, not just its middle band, has a product neighborhood.
Shrink it, if necessary, so that
\[
 d_{\mathrm{ann}}<
 \min\{c_P,\epsilon_{\mathrm{work}},\eps_{\mathrm{pair}}^{\mathrm{src}}\}.
\]
Equation \eqref{eq:pre-dbx-spectator-product} then defines
\(\widehat a_{e,u}^{\mathrm{sp}}\) from the pre-edge product coordinates.
The endpoint's own \(b_u,c_u\) fills the realized disk compatibly.  No
post-extension atlas or edge representative occurs.
\end{proof}

For the later inner/outer proof, keep disk and surface types separate:
\begin{align}
 C_+^D&=\{r<r_+\},&
 \mathcal A_{-+}^D&=\{r_-<r<r_+\},\notag\\
 C_+^S&=\zeta_{\mathrm{mod}}(C_+^D),&
 \mathcal W_-&=\{y\in\mathsf W:\bar r(y)>r_-\},\notag\\
 \mathcal A_{-+}^S&=\zeta_{\mathrm{mod}}(\mathcal A_{-+}^D)
  =(C_+^S\cap\mathsf W)\cap\mathcal W_-.
 \label{eq:inner-outer-regions}
\end{align}
Then
\begin{equation}
 \mathsf W=(C_+^S\cap\mathsf W)\cup\mathcal W_-,
 \qquad
 D^2=C_+^D\cup
 \zeta_{\mathrm{mod}}^{-1}(\mathcal W_-\cap Z_{\mathrm{mod}}).
 \label{eq:relative-band-covers}
\end{equation}

\subsection{A relative decorated-ball extension}

\begin{lemma}[Relative decorated-ball extension]
\label{lem:decorated-ball-extension}
Let \(B_0,B_1\) be compact smooth oriented \(3\)-balls.  Suppose
\[
 \partial B_i=P_i\cup\bigcup_{\lambda\in\Lambda}Z_{i,\lambda},
 \qquad
 P_i\cap Z_{i,\lambda}=A_{i,\lambda},
\]
where \(\Lambda\) is finite, the \(Z_{i,\lambda}\) are pairwise disjoint
closed disks, and the \(A_{i,\lambda}\) are full closed annuli.  Let
\(a:P_0\to P_1\) be an orientation-preserving diffeomorphism in compatible
product charts, preserving every label and acting as the identity in a
specified interface product coordinate.

Assume full positive disk charts
\(\zeta_{i,\lambda}:D^2\to Z_{i,\lambda}\) have already been chosen and
that, on every complete overlap annulus,
\begin{equation}
 \zeta_{1,\lambda}^{-1}a\zeta_{0,\lambda}(r,[\theta])
 =(r,[f_\lambda(\theta)]),\qquad
 f_\lambda(\theta+1)=f_\lambda(\theta)+1,\quad
 f_\lambda'(\theta)>0.
 \label{eq:dbx-annular-normal-form}
\end{equation}
Then \(a\) extends to an orientation-preserving diffeomorphism
\[
 T:B_0\longrightarrow B_1
\]
with the prescribed full outer-collar formula and identity interface
product germ, hence with equality of every jet on every protected germ.
If a finite union of compatible open product germs is specified in
advance, the extension may be required to agree there as well.
\end{lemma}

\paragraph{Radial interpolation and collar extension.}
The annular normal form is first interpolated radially between the prescribed
boundary map and the identity near each disk centre. Endpoint flatness gives
agreement of all tangential and normal derivatives on the full annuli, after which collars and the doubled-ball
approximation extend the boundary map without disturbing the protected germs.
The full disk charts are therefore essential hypotheses of the extension.

\begin{proof}
Choose an endpoint-flat radial cutoff \(\eta\), equal to zero near the
center and one on a neighborhood of each full annulus, and set
\[
 f_{\lambda,s}(\theta)=(1-s)\theta+s f_\lambda(\theta).
\]
Since
\(
\partial_\theta f_{\lambda,s}
=(1-s)+sf_\lambda'(\theta)>0
\),
the formula
\[
 a_{Z,\lambda}\zeta_{0,\lambda}(r,[\theta])
 =\zeta_{1,\lambda}
   (r,[f_{\lambda,\eta(r)}(\theta)])
\]
is a positive disk diffeomorphism, coordinate identity near the center,
and equal to \(a\) on the entire annulus.  Endpoint flatness gives agreement
of all tangential and normal derivatives at both radial seams.  The disk maps and \(a\) therefore glue to
an orientation-preserving sphere diffeomorphism \(a_\partial\).

Choose inward collars extending all named normal coordinates.  On a full
outer subcollar use
\[
 c_0(x,t)\longmapsto c_1(a_\partial(x),t).
\]
After removing a smaller open collar, the remaining pieces are closed
balls.  Write
\(\psi_\partial:S^2\to S^2\) for the induced inner-sphere
homeomorphism.  In fixed ball coordinates extend it by the radial
Alexander formula \(ru\mapsto r\,\psi_\partial(u)\).
This gives a ball homeomorphism already smooth on a full boundary collar.

Double the balls and this map.  The double is smooth on a full two-sided
equatorial neighborhood.  Let \(K\) be the equator together with a closed
thinner two-sided collar containing all preassigned compatible germs.
The classical relative homeomorphism-approximation theorem, in the form
stated by Pardon \cite[Theorem~4.6]{Pardon2020}, applied to the doubled
homeomorphism itself, supplies a
diffeomorphism of the source double to the target double which equals the
doubled map on a neighborhood of \(K\).  The equator separates both
doubles, and the fixed product germ identifies the selected source and
target sides.  Connectedness puts the image of the selected source side
inside the selected target side; the same argument for the inverse proves
surjectivity onto it.  Restriction to their closures is the required
\(T\).  Its determinant is positive on the unchanged collar and hence on
the connected ball.
\end{proof}

\begin{remark}
The lemma requires the full disk charts and
\eqref{eq:dbx-annular-normal-form} as hypotheses.  It does not assert that
an arbitrary annulus map has that form in independently chosen disk
charts.  This is why \cref{prop:relative-band-data} precedes every
spectator extension.
\end{remark}

\subsection{Compatible core and full-disk collars}

On the product neighborhood from
\cref{prop:relative-band-data}, let
\[
 X_e^{\mathrm{fix}}=(q_e^{\mathrm{sp}})_*(\partial_t).
\]
Choose
\(\overline{G_e^{\mathrm{keep}}}\Subset G_e^{\mathrm{work}}\)
inside the existing spectator collar, with the keep set containing the
entire closed annular overlap through positive depth.  Extend the fixed
field to a strictly inward field \(Y_e\) near the filled disk and use a
cutoff \(\chi_e\) to form
\[
 X_e^{\mathrm{pre}}
 =\chi_eX_e^{\mathrm{fix}}+(1-\chi_e)Y_e .
\]
The inward cone is convex.  A uniform short-time flow \(\Phi_{e,t}\)
therefore defines, after shrinking \(d_e^c<d_{\mathrm{ann}}\),
\begin{equation}
 \iota_{e,\mathrm{loc}}^c(x,t)
 =\Phi_{e,t}(j_e^c(x)),
 \qquad (x,t)\in C_{r_1}^D\times[0,d_e^c).
 \label{eq:core-collar}
\end{equation}
Uniqueness of integral curves gives, on the complete annulus,
\begin{equation}
 \iota_{e,\mathrm{loc}}^c(x,t)
 =q_e^{\mathrm{sp}}(\zeta_{\mathrm{mod}}(x),t).
 \label{eq:core-collar-agreement}
\end{equation}
Because the cutoff is one on an open neighborhood of the closed annulus,
this is an all-jet equality at both boundary circles.

Let
\[
 \zeta_e^{\mathrm{sp},\sigma_e}:D^2\longrightarrow
 Z_e^{\mathrm{sp},\sigma_e},
 \qquad
 \zeta_{e,u}^{\mathrm{sp},\sigma_e}:D^2\longrightarrow
 Z_{e,u}^{\mathrm{sp},\sigma_e}
\]
denote the retained pre-extension positive disk charts, and set
\begin{equation}
 \gamma_{e,u}^{\mathrm{pre}}
 =\zeta_{e,u}^{\mathrm{sp},\sigma_e}
   (\zeta_e^{\mathrm{sp},\sigma_e})^{-1}.
 \label{eq:retained-preextension-disk-comparison}
\end{equation}
Write \(a_{e,u}^{\mathrm{sp}}\) for the boundary trace of
\(\widehat a_{e,u}^{\mathrm{sp}}\) on the open annular germ, and put
\[
 d_{\mathrm{seam}}=\min\{d_{\mathrm{ann}},d_e^c\}.
\]

\Needspace{4\baselineskip}
\begin{proposition}[Positive spectator disk comparison]
\label{prop:spectator-disk-comparison}
For every incidence \((e,u)\) there is a positive full-disk
diffeomorphism
\[
 \gamma_{e,u}^z:
 Z_e^{\mathrm{sp},\sigma_e}
 \longrightarrow Z_{e,u}^{\mathrm{sp},\sigma_e}
\]
which equals
\(
b_u\zeta_{\mathrm{mod}}(j_e^c)^{-1}
\)
on
\(\mathcal U_{\mathrm{in}}=j_e^c(C_+^D)\) and equals
\(\gamma_{e,u}^{\mathrm{pre}}\) on
\[
 \mathcal U_{\mathrm{out}}
 =Z_e^{\mathrm{sp},\sigma_e}
   \setminus j_e^c(C_{r_-}^D).
\]
On the whole protected annulus it agrees with
\(a_{e,u}^{\mathrm{sp}}\), and its product lift satisfies
\begin{equation}
 \widehat\gamma_{e,u}^z
 q_e^{\mathrm{sp}}(\zeta_{\mathrm{mod}}(x),t)
 =\widehat a_{e,u}^{\mathrm{sp}}
 q_e^{\mathrm{sp}}(\zeta_{\mathrm{mod}}(x),t)
 =q_u^P(\zeta_{\mathrm{mod}}(x),t)
 \label{eq:full-annulus-all-jets}
\end{equation}
on an open germ containing
\(\mathcal A_{01}^D\times[0,d_{\mathrm{seam}})\), with all jets.
If
\[
 \zeta_{e,u}^{z,\sigma_e}
 :=\gamma_{e,u}^z\zeta_e^{\mathrm{sp},\sigma_e},
\]
then the disk-extension transition is, on the entire overlap annulus,
\begin{equation}
 a_{e,u}^{\mathrm{sp}}
 =\zeta_{e,u}^{z,\sigma_e}
   (\zeta_e^{\mathrm{sp},\sigma_e})^{-1}.
 \label{eq:full-annulus-transition}
\end{equation}
\end{proposition}

\begin{proof}
Use the protected formula on \(j_e^c(C_+^D)\) and the retained
pre-extension disk
map outside \(j_e^c(C_{r_-}^D)\).  On their open overlap the operative
chain is
\begin{equation}
 \gamma_{e,u}^{\mathrm{pre}}j_e^c(x)
 =a_{e,u}^{\mathrm{sp}}j_e^c(x)
 =b_u\zeta_{\mathrm{mod}}(x)
 =\gamma_{e,u}^zj_e^c(x),
 \qquad x\in\mathcal A_{-+}^D.
 \label{eq:disk-comparison-seam}
\end{equation}
This is a restriction of the pre-extension product identity
\eqref{eq:pre-dbx-spectator-product}, together with
\eqref{eq:core-collar-agreement}, and therefore holds on an open germ,
not merely on its central circle.  The formulas glue to a positive local
diffeomorphism of the closed disk.  It is proper, has boundary degree
\(+1\), and hence has disk degree \(+1\).  A proper local diffeomorphism
of a disk of degree one is a one-sheeted covering, so it is a
diffeomorphism.

With endpoints included by continuity, the complete annular transition
is the exhaustive three-piece formula
\[
 \gamma_{e,u}^zj_e^c(x)=a_{e,u}^{\mathrm{sp}}j_e^c(x)=
 \begin{cases}
 b_u\zeta_{\mathrm{mod}}(x),&r_0\le r(x)\le r_-,\\
 b_u\zeta_{\mathrm{mod}}(x)=\gamma_{e,u}^{\mathrm{pre}}j_e^c(x),
   &r_-\le r(x)\le r_+,\\
 \gamma_{e,u}^{\mathrm{pre}}j_e^c(x),&r_+\le r(x)\le r_1.
 \end{cases}
\]
Each row preserves the same inward coordinate on a surrounding open
product germ.  This proves \eqref{eq:full-annulus-all-jets} on the
entire annulus, including every tangent, normal, and mixed derivative.
Equation \eqref{eq:full-annulus-transition} is then the definition of
\(\zeta_{e,u}^{z,\sigma_e}\) rewritten in invariantly typed form.
\end{proof}

\subsection{Five model collars and ten realized endpoint collars}

For a model edge \(e\), let
\(\mathcal O_{e,\mathrm{germ}}^{\mathrm{mod}}\) be the open union of the
following restricted product germs, and let
\(O_{0,e}^{\mathrm{mod}}\) contain the closures of their required base
loci:
\begin{enumerate}[label=\textup{(M\arabic*)}]
\item the core collar \(\iota_{e,\mathrm{loc}}^c\) only over the disk-typed
inner region \(C_+^D\);
\item the outer-work collar \(q_e^{\mathrm{sp}}\) only over
\(\mathcal W_-=\{y\in\mathsf W:\bar r(y)>r_-\}\);
\item the complete protected product germ on
\(P_e^{\mathrm{sp}}\cap Z_e^{\mathrm{sp},\sigma_e}\);
\item the retained outer and residual disk annuli; and
\item every protected, pairing, ribbon, spoke, detector, and signed
\(E_e\) internal-interface product germ.
\end{enumerate}
The restrictions in (M1)--(M2) are part of the construction: neither
prescription is extended across the other one's exclusive region.

There are three potentially nonformal overlaps.  Core/protected is the
region \(r_0\le r<r_+\), where
\eqref{eq:core-collar-agreement} and
\eqref{eq:full-annulus-all-jets} give literal equality with all jets.
Core/outer is \(\mathcal A_{-+}^D\), hence is a restriction of that same
identity.  Protected/outer is the part of the complete annulus with
\(\bar r>r_-\), where both entries are the identical retained
\(q_e^{\mathrm{sp}}\)-product germ.  The same equation holds on every
triple intersection.  All other intersections are unchanged product
intersections from the common endpoint chart-and-marking system.  Thus the compatibility
check precedes the collar extension.  Choose
\begin{equation}
 O_{0,e}^{\mathrm{mod}}\Subset O_{1,e}^{\mathrm{mod}}
 \Subset\mathcal O_{e,\mathrm{germ}}^{\mathrm{mod}}
 \subset\partial L_{e,\mathrm{mod}}^{\mathrm{sp}}.
 \label{eq:model-multigerm-collar-domains}
\end{equation}
The formulas glue to one local collar on \(O_{1,e}^{\mathrm{mod}}\).
Apply \cref{lem:multigerm-relative-collar-extension} once for each of the
five model edges to obtain
\begin{equation}
 d_e:\partial L_{e,\mathrm{mod}}^{\mathrm{sp}}\times[0,b_e^{\mathrm{mod}})
 \hookrightarrow L_{e,\mathrm{mod}}^{\mathrm{sp}},
 \qquad b_e^{\mathrm{mod}}>0.
 \label{eq:five-model-collars}
\end{equation}

For each realized endpoint incidence \((e,u)\), take the corresponding restricted
list: the \(z_u\)-core germ only for \(x\in C_+^D\), the realized
outer-work germ only over \(\mathcal W_-\), both complete endpoint disk
charts, the complete realized protected-annulus product germ, and all
retained protected and interface product germs.  The endpoint's fixed
\(b_u,c_u\), together with the three-piece identity in the proof of
\cref{prop:spectator-disk-comparison}, carries each of the three model
overlap checks literally to this realized list.  Hence no endpoint-specific
compatibility is delegated to the collar lemma.  Let \(O_{0,e,u}\) contain
the closures of the corresponding required base loci.  If
\(\mathcal O_{e,u,\mathrm{germ}}\) is the open union of these germs, choose
\begin{equation}
 O_{0,e,u}\Subset O_{1,e,u}
 \Subset\mathcal O_{e,u,\mathrm{germ}}
 \subset\partial L_{e,u}^{\mathrm{sp}},
 \label{eq:physical-multigerm-collar-domains}
\end{equation}
and apply the lemma once for each of the ten realized endpoint incidences to obtain
\begin{equation}
 d_{e,u}:\partial L_{e,u}^{\mathrm{sp}}\times[0,b_{e,u})
 \hookrightarrow L_{e,u}^{\mathrm{sp}},
 \qquad b_{e,u}>0.
 \label{eq:ten-physical-collars}
\end{equation}
Thus there are exactly five positive model widths and ten positive
realized endpoint widths.

Put
\begin{equation}
 \iota_e^{\mathrm{core}}(x,t)=d_e(j_e^c(x),t),
 \qquad (x,t)\in
 C_{r_1}^D\times[0,b_e^{\mathrm{mod}}).
 \label{eq:extended-model-core-collar}
\end{equation}
The realized relative extension has the corresponding core-collar identity
\begin{equation}
 d_{e,u}(\gamma_{e,u}^zj_e^c(x),t)=z_u(x,t),
 \qquad x\in C_+^D,\quad
 0\le t<\min\{b_e^{\mathrm{mod}},b_{e,u}\}.
 \label{eq:physical-core-collar-crosswalk}
\end{equation}
Only now choose the common comparison depth
\begin{equation}
 u_{\mathrm{cmp}}
 =\frac12\min\left\{
 u_0,\eps_{\mathrm{pair}}^{\mathrm{src}},
 \epsilon_{\mathrm{work}},d_{\mathrm{ann}},
 \min_e d_e^c,\min_e b_e^{\mathrm{mod}},
 \min_{(e,u)\in\mathcal I_{10}}b_{e,u}
 \right\}>0 .
 \label{eq:comparison-depth}
\end{equation}
In particular \(u_{\mathrm{cmp}}<u_0\).  This restricts comparison
equations; it neither changes the domain nor redefines any already fixed
vertex map \(z_v\), written \(z_u\) when \(u\) is the endpoint in
question.

\Needspace{4\baselineskip}
\begin{proposition}[Selected spectator endpoint chart-and-marking systems]
\label{prop:spectator-atlases}
For every \((e,u)\in\mathcal I_{10}\), relative multi-disk extension gives
\[
 \Xi^z_{e,u}:L_{e,\mathrm{mod}}^{\mathrm{sp}}
 \longrightarrow L_{e,u}^{\mathrm{sp}}
\]
such that
\begin{align}
 \Xi^z_{e,u}\iota_e^{\mathrm{core}}(x,t)
 &=z_u(x,t),
 &&x\in C_+^D,\quad 0\le t\le u_{\mathrm{cmp}},
 \label{eq:xiz-inner}\\
 \Xi^z_{e,u}q_e^{\mathrm{sp}}(y,t)
 &=q_u^P(y,t),
 &&(y,t)\in
 \mathcal D_{e,\mathrm{sp}}^{\mathrm{work}}
 \cap(\mathcal W_-\times[0,u_{\mathrm{cmp}}]).
 \label{eq:xiz-outer}
\end{align}
It also preserves, on their complete product domains, the exact labeled
list
\[
 E,\quad \tau_p,\quad\tau_q,\quad s^-,\quad s^+,\quad
 s_s,\quad c_s,\quad b_{\delta_E},
\]
together with both complementary-disk annuli and the signed
internal-interface bicollar.  These are respectively the global cut
attachment, the two common spectator arcs, both spoke pieces, the
spectator branch, and the retained ribbon/detector germs.  For every such
label \(a\),
\begin{equation}
 \Xi^z_{e,u}\chi_{a,e}^{\mathrm{sp}}(x,\mathbf t)
 =\chi_{a,e,u}^{\mathrm{sp}}(x,\mathbf t)
 \quad\text{with all tangent, normal, framing, and jet data.}
 \label{eq:xiz-mark}
\end{equation}
\end{proposition}

\begin{proof}
Apply \cref{lem:decorated-ball-extension} to the two full disk charts and
the already compatible model and realized collars.  Its full-annulus
hypothesis is precisely \eqref{eq:full-annulus-all-jets}; the smaller
split seam is not substituted for it.  Take as relative fixed set the
union of the inner-core and outer-work prescriptions.  They agree on the
whole surface overlap, while every protected-annulus transition has
already been checked on its complete domain.  The relative conclusion
gives \eqref{eq:xiz-inner}--\eqref{eq:xiz-mark}.
\end{proof}

\begin{warning}[The outer equation is deliberately restricted]
Equation \eqref{eq:xiz-outer} is not asserted on all of
\(\mathcal D_{e,\mathrm{sp}}^{\mathrm{work}}\).  Extending it beyond
\(\mathcal W_-\) while retaining \eqref{eq:xiz-inner} would prescribe two
independent model inputs at points where their equality has not been
proved, contradicting injectivity in general.
\end{warning}

\subsection{Selection of the edge representatives}

All ten incidence-labeled maps \(\Xi^z_{e,u}\) are now fixed, so the local
representatives can be chosen alongside already specified spectator maps.
For each edge, apply the recipe of \cref{prop:nat-schema}.  The maps
\(\Xi\) supply the spectator interface germ; the local extension recipe
still depends only on its independently constructed endpoint data.

\begin{proposition}[Selected naturalized representatives]
\label{prop:fresh-naturalized-representatives}
For every \(e:v\to w\), there is an orientation-preserving
\[
 \widehat\phi_e^{\mathrm{cut}}:
 B_{e,v}^{\mathrm{LS}}\longrightarrow B_{e,w}^{\mathrm{LS}}
\]
in the positive \(B0/B5\) relative class fixed by
\cref{thm:positive-two-site-input}.  It satisfies
\begin{equation}
 \widehat\phi_e^{\mathrm{cut}}
   \widehat q_{e,v}^{\mathrm{LS}}
 =\widehat q_{e,w}^{\mathrm{LS}},
 \qquad
 \widehat\phi_e^{\mathrm{cut}}(z,t)=(z,t)
 \label{eq:nat-edgewise}
\end{equation}
on the complete local shallow domain and signed local-interface germ,
respectively, with all jets.  It also intertwines separately the two source
active pairings with their corresponding target pairings on the full
half-collars.

On the common signed internal bicollar \(c_{e,u}^{E_e}\), the local and
spectator conjugates have the same product germ:
\begin{align}
 \widehat\Theta_{e,w}\widehat\phi_e^{\mathrm{cut}}
 \widehat\Theta_{e,v}^{-1}
  c_{e,v}^{E_e}(z,t)
 &=c_{e,w}^{E_e}(z,t),\notag\\
 \Xi^z_{e,w}(\Xi^z_{e,v})^{-1}
  c_{e,v}^{E_e}(z,t)
 &=c_{e,w}^{E_e}(z,t),
 \qquad |t|<\eta ,
 \label{eq:internal-bicollar-germs}
\end{align}
and every derivative of either coordinate expression minus \((z,t)\)
vanishes.
\end{proposition}

\begin{proof}
The compatible-extension schema already fixed the boundary transition and its
relative class without choosing an edge map.  Instantiate its
collar-extension, radial-homeomorphism, and relative-smoothing recipe now.
The common template \eqref{eq:nat-template} makes the full boundary collar
formula leave the inward variable unchanged.  Hence the shallow identity
in \eqref{eq:nat-edgewise} follows from \eqref{eq:qhat}; the two pairing
equations follow independently from the two abstract pairing
intertwinings of \cref{prop:native-atlases}.  The signed interface formula
is coordinate identity by construction.

Conjugating that interface formula by the endpoint maps
\(\widehat\Theta_{e,u}\) gives the first row of
\eqref{eq:internal-bicollar-germs}.  The second is
\eqref{eq:xiz-mark} for the signed spectator interface.  Both statements
hold on open product germs, so equality includes every jet and not just
the central disk.
\end{proof}

The closed local and spectator halves meet along the internal disk
\(E_{e,u}\), whereas \eqref{eq:internal-bicollar-germs} compares their
extensions on a signed open bicollar.  Here \(E_{e,u}\) is the internal
local/spectator interface and is not the global exterior disk \(E\).
The next section uses these two already constructed germs to define the
cut map.  Old regions and unused handles do not belong to either cut-ball
half and acquire an identity map only after quotient descent and global
extension.