\section{Common charts and endpoint markings}
\label{sec:endpoint-atlases}

The vertex data in \eqref{eq:constructed-vertex-package} are now fixed.
An endpoint chart-and-marking system gives a local coordinate model in which
incident edge maps can later be compared.  Three common pair-type footprints
and five rounded local models supply the shared geometry; ten
incidence-labelled endpoint-native atlases retain which endpoint each chart
describes.  These charts are chosen before the edge representatives, so none
can be tailored to an equation involving an unselected cut map.

For each pair type, the common disk must contain the active footprints from
every incidence of that type while avoiding their spectator markings.  At the exceptional
\(AC\starv\) ribbon it must also contain a reserved neighborhood for the
longitude correction, called the \emph{exceptional-ribbon reserve} below.
The rounded model over this disk gives compatible local coordinates at both
ends.  The endpoint charts and extensions are chosen before an edge
representative; the subsequent boundary isotopy, collar extension, and
smoothing place that representative in the prescribed relative class.

\subsection{Incidence-labelled support and marking footprints}

Let
\begin{equation}
 \mathcal I_{10}=\{(e,u):e\in E_5,\ u\text{ is an endpoint of }e\}.
 \label{eq:ten-endpoint-incidences}
\end{equation}
The incidence label is retained even if two atlases later have the same
underlying value.  For each \((e,u)\in\mathcal I_{10}\), distinguish the
following three objects.

\begin{enumerate}[label=\textup{(\alph*)},leftmargin=2.6em]
\item The \emph{named native carrier}
\(\mathcal S_{e,u}^{\mathrm{nat}}\): the mover meridian tube, the slide
band, the carrier longitude and push-off, the listed smoothing boxes, and
the direct-dual ribbons belonging to that typed local slide.  For the two
\(AC\starv\) incidences it also contains the complete exceptional-ribbon
reserve and its product sweep.

\item The compact \emph{active base footprint}
\(G_{e,u}^{\mathrm{mc}}\subset\partial K_u\), prescribed by the endpoint
role of the later transported named carrier.

\item The \emph{actual spectator-marking footprint}
\(\mathfrak M_{e,u}^{\mathrm{sp}}\subset\partial K_u\), including its
complementary disk, branch, full collar, ribbon, spoke, pairing,
eight-letter boundary word, detector, and protected-annulus footprint.
\end{enumerate}

The first object is a native three-dimensional set and is not yet a
subset of a realized \(c_u\)-chart.  It is also not asserted to be the
full support of any eventual representative.  The latter two are compact
subsets of the same concrete boundary sphere, so their separation is
well typed.  Pull them back before choosing a model footprint:
\begin{equation}
 \overline G_{e,u}^{\mathrm{mc}}=b_u^{-1}(G_{e,u}^{\mathrm{mc}}),
 \qquad
 \overline{\mathfrak M}_{e,u}^{\mathrm{sp}}
 =b_u^{-1}(\mathfrak M_{e,u}^{\mathrm{sp}})\subset S.
\label{eq:pulled-back-incidence-footprints}
\end{equation}

There is one deliberate exposition-order exception.  The
exceptional-ribbon reserve is a pre-edge choice: its row-typed forbidden sets,
positive clearances, swept strip, and the exceptional-ribbon reserve disk
\(A_{jk}^{\mathrm{ear}}\) are constructed using only the fixed vertex maps
and native carrier data.  The coordinate construction is printed at the
start of \cref{sec:naturalization}, where the exceptional ribbon is used,
but it is logically prior to the common-disk choice below.  In particular,
that construction does not consume an endpoint chart-and-marking system, an edge
representative, or the disk \(A_{jk}\).

The support scales belong to the same pre-edge choice.  Let
\(d_{\mathrm{ret}}>0\) be the minimum of the finitely many retained
positive product-germ depths.  The germs and global collars were frozen
with depth beyond \(3h\); after restricting them to one common depth,
choose \(d_{\mathrm{prod}}\) with \(h<d_{\mathrm{prod}}<2h\).  Once the
positive reserve clearances have been formed, make the simultaneous final
pre-edge scale choice in \eqref{eq:ear-clearance-scales}.  The resulting
values satisfy
\begin{equation}
 3c_P<n_0<n_1<h_0<h<d_{\mathrm{prod}}<2h<3h
 <\min\{b_{\mathrm{col}},d_{\mathrm{germ}},d_{\mathrm{ret}}\}.
 \label{eq:exceptional-depth-chain}
\end{equation}
Thus every scale used by the coordinate construction at the start of
\cref{sec:naturalization} is fixed before the three common disks.  All
earlier occurrences of these widths, including \(u_0=c_P/2\), mean their
final restricted values; no vertex witness is altered after it is declared
final.

\begin{lemma}[Three common pair-type disks]
\label{lem:three-common-footprint-disks}
There are smooth rounded disks
\(A_{ij},A_{ik},A_{jk}\Subset\mathsf W\), and a smaller closed rounded
disk \(A_{jk}^{\mathrm{ear}}\), such that
\begin{equation}
\begin{gathered}
 A_{jk}^{\mathrm{ear}}\Subset\Int A_{jk},\\
 \bigcup_{(e,u)\in\mathcal I_{pq}}
  \overline G_{e,u}^{\mathrm{mc}}
 \Subset\Int A_{pq}
 \Subset
 \mathsf W\setminus
 \bigcup_{(e,u)\in\mathcal I_{pq}}
  \overline{\mathfrak M}_{e,u}^{\mathrm{sp}},
 \qquad pq\in\{ij,ik,jk\},
\end{gathered}
\label{eq:common-footprint-containments}
\end{equation}
where \(\mathcal I_{pq}\) consists of the incidences having model pair
\((p,q)\).  The reserve disk contains, with margin, both
\(AC\starv\) exceptional reserves.
\end{lemma}

\begin{proof}
For the \(j,k\) pair, the exceptional reserve is enlarged without rechoice.

For a fixed pair \((p,q)\), take the finite union of the complete active
branch subtrees and the terminal portions of their spokes.  Join its two
active components by an arc in the annular work collar.  Use respectively
the gaps between the \(i,j\) boundary words, the cyclic wrap between the
\(k,i\) words, and the gap between the \(j,k\) words.  The connector
misses the complete spectator branch.  The resulting spine is a tree,
so a sufficiently small rounded regular neighborhood is a disk.

For the \(j,k\) pair, begin with the already chosen exceptional reserve
tree and its rounded disk \(A_{jk}^{\mathrm{ear}}\), then attach the
remaining finitely many prescribed subtrees by tree arcs; no cycle is
introduced.  The pre-edge participant/forbidden construction displayed in
\cref{tab:exceptional-ear-participants,eq:ear-boundary-clearance} and the
margin \eqref{eq:ear-model-disk-construction} make the finite active union
disjoint from the finite union of the actual spectator markings.  Enlarging
the tree neighborhood, rather than rechoosing the reserve disk, gives
\(A_{jk}^{\mathrm{ear}}\Subset\Int A_{jk}\).
Normality of the surface and compactness then give the two strict
containments in \eqref{eq:common-footprint-containments} and a positive
margin.  The choices for the three pair types are independent; the disks
are not required to be mutually disjoint.
\end{proof}

For an incidence, put \(A(e)=A_{pq}\) according to its model pair and
\begin{equation}
 A_{e,u}=b_u(A(e)).
 \label{eq:physical-incidence-disk}
\end{equation}
The full role table is
\begin{equation}
\begin{array}{c|c|c|c|c}
\text{incidence}&A(e)&\text{mover/carrier}&\text{spectator}
 &\text{exceptional-ribbon reserve}\\
\hline
(0B,0)&A_{jk}&j_0,k_0&i_0&--\\
(0B,B)&A_{jk}&J,k_0&i_0&--\\
(B\starv,B)&A_{ij}&i_0,J&k_0&--\\
(B\starv,\starv)&A_{ij}&I_\star,J&k_0&--\\
(0A,0)&A_{ij}&i_0,j_0&k_0&--\\
(0A,A)&A_{ij}&I_A,j_0&k_0&--\\
(AAC,A)&A_{ik}&I_A,k_0&j_0&--\\
(AAC,AC)&A_{ik}&I_\star,k_0&j_0&--\\
(AC\starv,AC)&A_{jk}&j_0,k_0&I_\star&inside\\
(AC\starv,\starv)&A_{jk}&J,k_0&I_\star&inside.
\end{array}
\label{tab:ten-endpoint-roles}
\end{equation}
% Internal search aliases retained for repository history:
% 0B@0, 0B@B, Bstar@B, Bstar@star, 0A@0, 0A@A,
% AAC@A, AAC@AC, ACstar@AC, ACstar@star.
Every row separately satisfies
\begin{equation}
 \overline G_{e,u}^{\mathrm{mc}}\Subset\Int A(e),\qquad
 G_{e,u}^{\mathrm{mc}}\Subset
 \Int_{\partial K_u}A_{e,u},\qquad
 \operatorname{dist}(A_{e,u},\mathfrak M_{e,u}^{\mathrm{sp}})>0.
 \label{eq:ten-own-incidence-clearances}
\end{equation}
These are own-incidence statements realized by three common finite-union
choices.  They do not claim that \(A_{e,u}\) avoids spectator markings
belonging to unrelated pair types.

Retain the source working domain
\begin{equation}
 D^{\mathrm{work}}
 =\mathsf W\times[0,\epsilon_{\mathrm{work}}),
 \qquad 0<\epsilon_{\mathrm{work}}<c_P<h_0.
 \label{eq:source-working-domain}
\end{equation}
This width is part of the pre-edge data.  It is not a later comparison,
pairing, or finite-compatibility depth.

\begin{lemma}[Exceptional-ribbon reserve clearance]
\label{lem:exceptional-ear-clearance}
At \(u=AC,\starv\), the full exceptional
\(AC\starv\) longitude-correction sweep lies in the rounded endpoint-specific
local ball based on \(A_{jk}\) and misses its side and cap.
\end{lemma}

\begin{proof}
The surface margin separates the sweep from the local ball's side, and
the product-depth bound separates it from the cap.  Both bounds use the
same vertex collar.  This containment is established before an edge
representative is selected; it does not construct the corrected row or
define a cross-endpoint edge map.

The direct vertex specialization gives, on the reserve disk,
\begin{equation}
 b_u^\partial|_{A_{jk}^{\mathrm{ear}}}
 =b_u|_{A_{jk}^{\mathrm{ear}}}.
 \label{eq:ear-vertex-binding}
\end{equation}
The already fixed collar \(c_u\) supplies the common product coordinate
there through depth \(2h\); no second ``vertex'' collar is introduced.
Recall the pre-edge scales \(n_0,n_1,d_{\mathrm{prod}}\) and
\(d_{\mathrm{ret}}\) fixed in \eqref{eq:exceptional-depth-chain}.
The closed sweep has base in
\(b_u(A_{jk}^{\mathrm{ear}})\) and depth at most \(n_1\).
The surface margin
\(b_u(A_{jk}^{\mathrm{ear}})\Subset\Int b_u(A_{jk})\), the common product
coordinate \eqref{eq:ear-vertex-binding}, and \(n_1<h<2h\) keep it away
from both the side over \(\partial A_{jk}\) and the cap at depth \(2h\).

The boundary and ambient separations used here are the finite positive
minima \eqref{eq:ear-boundary-clearance} and
\eqref{eq:ear-ambient-clearance}; the actual side/cap identification is
\eqref{eq:ear-physical-side-and-cap}.
\end{proof}

The three atlas layers now have separate jobs.  The common pair-type disks
provide one base region for all incidences with the same local pair; the
rounded local model turns that region into a common protected three-ball; and
the ten endpoint-native systems retain which edge endpoint is being described.
The first two layers make comparison possible, but they do not select an edge
map.  The endpoint-native layer is therefore the last chart datum before the
relative extension theorem, not a hidden transport rule between vertices.

\subsection{The common rounded local model}

For each edge form, in the abstract collared ball,
\begin{equation}
 K_{e,\mathrm{mod}}^{\mathrm{loc}}
 =\operatorname{Round}\bigl(A(e)\times[0,2h]\bigr),
 \label{eq:rounded-local-model}
\end{equation}
using one rounding profile at both endpoints.  Its internal interface is
\begin{equation}
 E_{e,\mathrm{mod}}
 =\operatorname{Round}\bigl(
 A(e)\times\{2h\}\cup\partial A(e)\times[0,2h]\bigr).
 \label{eq:edge-local-interface}
\end{equation}
This disk is the edge-local interface \(E_e\); it is not the global
exterior disk \(E\).  The common rounding fixes a signed bicollar
\begin{equation}
 c_{e,\mathrm{mod}}:D^2\times(-h_0,h_0)
 \longrightarrow B^3_{\mathrm{mod}},
 \label{eq:common-signed-interface-collar}
\end{equation}
negative on the local side and positive on the spectator side.

Collapse \(E_e\) to the central fat vertex and retain the two active
branches.  For the model pair \((p,q)\) the resulting local fat tree is
\begin{equation}
 E_e\xrightarrow{s_p}p^-\xrightarrow{c_p}p^+,\qquad
 E_e\xrightarrow{s_q}q^-\xrightarrow{c_q}q^+.
 \label{eq:local-two-branch-tree}
\end{equation}
Give \(E_e,p^-,q^-\) four sectors and \(p^+,q^+\) two sectors.  Its
regular neighborhood is a disk, and the complement in the local boundary
sphere is one disk \(Z_e^{\mathrm{loc}}\).  With
\begin{equation}
 B_r=f_{E_e,r}^{-1}r_{s_r}f_{r^-,1}^{-1}r_{c_r}
 f_{r^+}^{-1}l_{c_r}^{-1}f_{r^-,0}^{-1}l_{s_r}^{-1},
 \label{eq:eight-letter-branch-word}
\end{equation}
the oriented complementary boundary is \(B_pB_q\).  For \(A_{ik}\), the
connector crosses the cyclic wrap gap, so this word is read from the
start of \(B_i\); \(B_kB_i\) is the same cyclic word after an
eight-letter rotation.

Relabelling the standard two-site cellulation by
\begin{equation}
 E_\gamma\mapsto E_e,\qquad
 1^\pm\mapsto p^\pm,\qquad
 2^\pm\mapsto q^\pm
 \label{eq:two-site-cell-relabeling}
\end{equation}
maps every cell and cyclic boundary word, not merely the cell counts.
The independent cell count is
\begin{equation}
\begin{aligned}
 V_{\mathrm{loc}}&=2(4+4+2+4+2)=32,\\
 E_{\mathrm{loc}}&=3(4+4+2+4+2)+2\cdot4=56,\\
 F_{\mathrm{loc}}&=5+(4+4+2+4+2)+4+1=26.
\end{aligned}
\label{eq:local-model-census}
\end{equation}
Choose a positive chart
\begin{equation}
 \zeta_e^{\mathrm{loc}}:D^2\longrightarrow Z_e^{\mathrm{loc}}
 \label{eq:local-complementary-disk-chart}
\end{equation}
whose full outer annulus is radial and agrees with all sixteen indexed
edge parameters.  It is common to both endpoints and is chosen before
their endpoint-specific charts.

\subsection{The common spectator model and signed cover}

Let \(L_{e,\mathrm{mod}}^{\mathrm{sp}}\) be the closure of the complement
of \(K_{e,\mathrm{mod}}^{\mathrm{loc}}\) in the model ball.  Its protected
fat graph has vertex disks
\begin{equation}
 E,\quad E_e,\quad s^-,\quad s^+,
 \label{eq:spectator-fat-vertices}
\end{equation}
and edges
\begin{equation}
 \tau_p,\tau_q:E\to E_e,\qquad
 s_s:E\to s^-,\qquad c_s:s^-\to s^+.
 \label{eq:spectator-fat-edges}
\end{equation}
At \(E\) use the inherited rotation \((i,j,k)\), replacing each active
spoke by its continuation \(\tau_r\); at \(E_e\) use
\((\tau_p,\tau_q)\); at \(s^-\) use \((s_s,c_s)\); and at \(s^+\) use
the one-element rotation.  If \(\alpha_{\mathrm{he}}\) exchanges the two
half-edges of every edge and \(\sigma_{\mathrm{rot}}\) is this rotation,
then the boundary permutation
\(\beta_{\mathrm{bd}}=\sigma_{\mathrm{rot}}\alpha_{\mathrm{he}}\) has
the following two orbits:
\begingroup
\setlength{\arraycolsep}{2pt}
\begin{equation}
\begin{array}{c|l|l}
e&\text{six-half-edge orbit}&\text{two-half-edge orbit}\\
\hline
0B,AC\starv&
\bigl((c_i,i^+),(s_i,i^-),(\tau_j,E),(\tau_k,E_e),(s_i,E),(c_i,i^-)\bigr)&
\bigl((\tau_j,E_e),(\tau_k,E)\bigr)\\
B\starv,0A&
\bigl((c_k,k^+),(s_k,k^-),(\tau_i,E),(\tau_j,E_e),(s_k,E),(c_k,k^-)\bigr)&
\bigl((\tau_i,E_e),(\tau_j,E)\bigr)\\
AAC&
\bigl((c_j,j^+),(s_j,j^-),(\tau_k,E),(\tau_i,E_e),(s_j,E),(c_j,j^-)\bigr)&
\bigl((\tau_i,E),(\tau_k,E_e)\bigr).
\end{array}
\label{tab:spectator-boundary-orbits}
\end{equation}
\endgroup
% Internal @-notation aliases for the two boundary orbits remain recorded in
% evidence/publication_normalization/P2_PUBLIC_INCIDENCE_ALIAS_CROSSWALK.tsv.
Thus the regular neighborhood \(P_e^{\mathrm{sp}}\) of the rank-one fat
graph is an annulus and its complement consists of exactly two disks
\[
 Z_e^{\mathrm{sp},+},\qquad Z_e^{\mathrm{sp},-}.
\]
Their signs are fixed by the two sides of the oriented global word
\(B_iB_jB_k\) and are never exchanged at an endpoint.  Expanding the
half-edge intervals gives the check
\begin{equation}
\begin{aligned}
V_{\mathrm{sp}}&=2(6+4+4+2)=32,\\
E_{\mathrm{sp}}&=3(6+4+4+2)+2\cdot4=56,\\
F_{\mathrm{sp}}&=4+(6+4+4+2)+4+2=26.
\end{aligned}
\label{eq:spectator-model-census}
\end{equation}
At this stage the two labels and their full radial annular germs are
fixed, while the inner disk fills remain available for the relative
band choice in \cref{sec:relative-extensions}.  This is essential at the
polar centers: a boundary-circle agreement alone does not determine a
smooth positive disk chart.

At an endpoint \(u\), bind the corresponding endpoint-specific halves by
\begin{equation}
 K_{e,u}^{\mathrm{loc}}
 :=\operatorname{Round}\bigl(c_u(A_{e,u}\times[0,2h])\bigr),
 \qquad
 L_{e,u}^{\mathrm{sp}}
 :=\overline{K_u\setminus K_{e,u}^{\mathrm{loc}}}.
 \label{eq:physical-local-spectator-balls}
\end{equation}
They meet exactly in the rounded realized copy of \(E_{e,\mathrm{mod}}\).

The common rounding also yields an endpoint-independent smooth separator
\begin{equation}
 \rho_e:D^{\mathrm{work}}\longrightarrow\R
 \label{eq:signed-rounded-separator}
\end{equation}
which is negative on the local side, zero exactly on the rounded
interface, and positive on the spectator side.  It is the signed normal
coordinate in \eqref{eq:common-signed-interface-collar} and is extended
by negative and positive constants of absolute value at least \(h_0\)
away from that collar.  Define the relatively closed halves
\begin{equation}
\mathcal D_{e,\mathrm{loc}}^{\mathrm{work}}=\{\rho_e\leq0\},\qquad
\mathcal D_{e,\mathrm{sp}}^{\mathrm{work}}=\{\rho_e\geq0\}.
\label{eq:closed-signed-halves}
\end{equation}
Then
\begin{equation}
D^{\mathrm{work}}
=\mathcal D_{e,\mathrm{loc}}^{\mathrm{work}}
 \cup\mathcal D_{e,\mathrm{sp}}^{\mathrm{work}},\qquad
\mathcal D_{e,\mathrm{loc}}^{\mathrm{work}}
 \cap\mathcal D_{e,\mathrm{sp}}^{\mathrm{work}}
=\rho_e^{-1}(0),
\label{eq:exact-signed-cover}
\end{equation}
and, at either endpoint,
\begin{equation}
 q_u^P(\mathcal D_{e,\mathrm{loc}}^{\mathrm{work}})
 \subset K_{e,u}^{\mathrm{loc}},\qquad
 q_u^P(\mathcal D_{e,\mathrm{sp}}^{\mathrm{work}})
 \subset L_{e,u}^{\mathrm{sp}}.
 \label{eq:physical-signed-half-containment}
\end{equation}
For a fixed \(0<\delta_e<h_0\), only the chart extensions have the open
domains
\begin{equation}
 \mathcal O_{e,\mathrm{loc}}=\{\rho_e<\delta_e\},\qquad
 \mathcal O_{e,\mathrm{sp}}=\{\rho_e>-\delta_e\}.
 \label{eq:open-signed-chart-domains}
\end{equation}
The closed halves in \eqref{eq:closed-signed-halves} are not silently
replaced by these larger open sets.

\subsection{Ten endpoint-native local chart systems}

For \(e:v\to w\), let \(B_{e,u}^{\mathrm{LS}}\) be the endpoint ball
furnished by \cref{thm:positive-two-site-input} after the typed
substitution in \cref{tab:five-source-edge-roles}.  Its indexed boundary
cellulation is the one relabelled in
\eqref{eq:two-site-cell-relabeling}.  The cellwise product charts and the
positive polar fill on its unique complementary disk give, independently
at the two endpoints,
\begin{equation}
 \widehat h_{e,u}:
 \partial K_{e,\mathrm{mod}}^{\mathrm{loc}}
 \overset{\cong}{\longrightarrow}
 \partial B_{e,u}^{\mathrm{LS}},
 \qquad (e,u)\in\mathcal I_{10}.
 \label{eq:ten-native-boundary-atlases}
\end{equation}
Every indexed face, ribbon, interface germ, and complete complementary
disk is included.  There is no source-to-target transition in the
definition of either endpoint chart-and-marking system.

Independently, push the rounded local model through the already fixed
vertex chart:
\begin{equation}
 G_{e,u}^{\mathrm{loc}}(x,t)=c_u(b_u(x),t)
 \quad\text{on }\operatorname{Round}(A(e)\times[0,2h]),
 \qquad
 g_{e,u}^{\mathrm{loc}}
 =G_{e,u}^{\mathrm{loc}}|_{\partial K_{e,\mathrm{mod}}^{\mathrm{loc}}}.
 \label{eq:ten-physical-local-atlases}
\end{equation}
The common rounding makes \(G_{e,u}^{\mathrm{loc}}\) an
orientation-preserving ball diffeomorphism onto
\(K_{e,u}^{\mathrm{loc}}\).  It is constructed from \(b_u,c_u\), not
from a desired edge equation.

Before any ball extension, choose for each edge one native collar width
and one compression
\begin{equation}
\ell_e>0,\qquad
\lambda_e:[0,h_0)\longrightarrow[0,\ell_e),\quad
\lambda_e(0)=0,\quad\lambda_e'(t)>0,\quad
\lambda_e(t)=t\text{ near }0,
\label{eq:common-endpoint-compression}
\end{equation}
with
\(\lambda_e([0,\epsilon_{\mathrm{work}}])\Subset[0,\ell_e)\).
The same \((\ell_e,\lambda_e)\) is used at both endpoints.  The actual
endpoint-only boundary identification is
\begin{equation}
 \widehat a_{e,u}^{\mathrm{loc}}
 =g_{e,u}^{\mathrm{loc}}(\widehat h_{e,u})^{-1}:
 \partial B_{e,u}^{\mathrm{LS}}
 \longrightarrow\partial K_{e,u}^{\mathrm{loc}}.
 \label{eq:endpoint-only-boundary-identification}
\end{equation}
In full disk charts it is the coordinate identity on the protected
region, the complete complementary disk, and their full annular overlap.

There is one exceptional endpoint decoration.  At the \(AC\starv\)
source, the native direct rows contain \(L_{ji},K_{ki}\), so the raw
carrier longitude is
\begin{equation}
 \widetilde K
 =K_{ki}\natural_{\beta_{AC\starv}^{\vee}}(-L_{ji})^+.
 \label{eq:raw-exceptional-longitude}
\end{equation}
At the target the directly constructed terminal row already contains
\(K_{kj}\).  The product-normal ear cancellation carries
\(\widetilde K\) to \(K_{kj}\), relative to its common tail and all
other marking germs.  By \cref{lem:exceptional-ear-clearance}, its whole
sweep stays in the target local ball.  Use the terminal time of this
isotopy only to decorate the target component of
\(\widehat a_{AC\starv,\starv}^{\mathrm{loc}}\).  It neither constructs
\(K_{kj}\) nor defines a cross-endpoint edge map, and no decoration is
made at the source.

\begin{lemma}[Relative local-ball extension]
\label{lem:relative-local-ball-extension}
For every incidence there is an orientation-preserving diffeomorphism
\begin{equation}
 \widehat\Theta_{e,u}:B_{e,u}^{\mathrm{LS}}
 \overset{\cong}{\longrightarrow}K_{e,u}^{\mathrm{loc}}
 \label{eq:ten-local-ball-extensions}
\end{equation}
extending \eqref{eq:endpoint-only-boundary-identification} and satisfying
\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:local-ball-outer-collar-formula}
\end{equation}
on the full prescribed outer collar.  On the signed internal interface
it is the coordinate identity, with all jets.
\end{lemma}

\begin{proof}
The boundary map is already a product map on mutually compatible open
collars of every protected face, the complete complementary disk, and
the signed internal interface.  Extend it by the displayed product
formula over a smaller boundary collar.  Removing that collar leaves two
smooth inner balls and an orientation-preserving sphere map that is
fixed on the induced protected disks.  Radial Alexander extension gives
a homeomorphism of the inner balls which is already the displayed smooth
product map on a full boundary collar.  The remaining step is a relative
smooth extension, and we record why the relative clause applies rather
than treating the Alexander formula as smooth at its cone point.

Double the two inner balls and the homeomorphism across that product
collar.  This gives a homeomorphism between smooth closed
three-manifolds which is already a diffeomorphism on a full open
two-sided neighborhood of the equator.  Let \(K\) be the equator together
with a closed thinner two-sided collar containing every preassigned
protected germ.  The classical relative homeomorphism-approximation theorem,
in the form stated by Pardon \cite[Theorem~4.6]{Pardon2020}, supplies a
diffeomorphism of the two doubles equal to the doubled homeomorphism on a
neighborhood of \(K\).

The equator separates each double into two connected components.  On its
fixed collar the approximating diffeomorphism is the side-preserving
product map, so the connected image of the selected source side lies in
the selected target side.  The same argument for the inverse gives the
reverse inclusion.  Restriction to the closures therefore gives the
required smooth inner-ball extension.  Its determinant is positive on
the unchanged collar and hence everywhere on the connected ball.

Gluing this extension to the original outer-collar formula produces
\(\widehat\Theta_{e,u}\); equality on an open product overlap gives
agreement of every derivative.  The exceptional target uses the
relative product isotopy above, whose support has already been proved to
miss the side and cap, as one of the protected germs held fixed by the
relative smoothing.
\end{proof}

Only after \(\widehat\Theta_{e,u}\) exists define the native shallow
embedding on the closed local half by
\begin{equation}
 \widehat q_{e,u}^{\mathrm{LS}}(x,t)
 =\bigl(\widehat h_{e,u}(x),\lambda_e(t)\bigr),
 \qquad
 (x,t)\in\mathcal D_{e,\mathrm{loc}}^{\mathrm{work}}.
 \label{eq:native-shallow-embedding}
\end{equation}
Because \(G_{e,u}^{\mathrm{loc}}=q_u^P\) on that half,
\eqref{eq:local-ball-outer-collar-formula} gives
\begin{equation}
 \boxed{
 \widehat\Theta_{e,u}^{-1}q_u^P
 =\widehat q_{e,u}^{\mathrm{LS}}
 \quad\text{on }\mathcal D_{e,\mathrm{loc}}^{\mathrm{work}}.}
 \label{eq:preselection-local-naturality}
\end{equation}
The ball extension therefore identifies the native shallow collar with the
already fixed vertex collar.  This equation follows from the extension;
it does not define either endpoint chart.

The realized image of the named native carrier is now well typed:
\begin{equation}
 \mathcal S_{e,u}^{\mathrm{phys}}
 :=\widehat\Theta_{e,u}(\mathcal S_{e,u}^{\mathrm{nat}})
 \subset c_u(A_{e,u}\times[0,h))
 \Subset K_{e,u}^{\mathrm{loc}}.
 \label{eq:transported-native-carrier}
\end{equation}
For the eight ordinary incidences it lies more sharply in
\(c_u(G_{e,u}^{\mathrm{mc}}\times[0,h_0])\); the two exceptional
incidences use \cref{lem:exceptional-ear-clearance}.  This is a statement
about the named carrier, not about arbitrary interior smoothing support
of a future representative.

Finally, and only after the endpoint data at both ends in
\eqref{eq:ten-local-ball-extensions}--\eqref{eq:preselection-local-naturality}
are in place, define the native boundary transition
\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:post-atlas-boundary-transition}
\end{equation}
Equation \eqref{eq:post-atlas-boundary-transition}, the ten atlases, and
the deterministic extension recipe form an \emph{uninstantiated
compatible-extension schema}.  They do not select or name a cut
representative.

\begin{remark}[What remains deliberately unfrozen]
\label{rem:endpoint-atlas-boundary}
The final spectator ball maps and the theorem's ten atlases
\(\Xi^z_{e,u}\) are not the maps
\(\widehat\Theta_{e,u}\).  Their two complementary disk charts must first
be chosen relative to actual overlap bands and compatible inward collars;
that construction occurs in \cref{sec:relative-extensions}.  Only after
all ten final endpoint atlases have been frozen may the compatible-extension
schema select the five cut representatives.  This preserves both the
vertex-first quantifier order and the distinction between the local
interface \(E_e\) and the global interface \(E\).
\end{remark}