\section{Vertex-first cut geometry}
\label{sec:vertex-geometry}

At a shared endpoint, composition requires one marked cut ball and one set
of collars for both incident edges.  We therefore construct the complete
vertex rows and their cut balls before choosing any edge map.  The same
vertex collars will later coordinate the local and spectator extensions.

Throughout this section the protected input \(\cT\) is fixed, but no edge
representative is chosen.  Every object is constructed once for each
\(v\in V\), directly from the row in
\eqref{tab:five-active-disk-rows}.  Every incident edge uses this same
vertex datum.

\subsection{Complete rows and relative ball identifications}

For \(v\in V\), cut the active sector along its three displayed row disks:
\begin{equation}
 K_v
 =U_{ijk}\mathbin{\backslash\!\backslash}
   (D_i^v\cup D_j^v\cup D_k^v).
 \label{eq:vertex-cut-ball}
\end{equation}

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[
  obj/.style={draw,rounded corners=5pt,minimum width=4.2cm,minimum height=2.05cm},
  outputbox/.style={draw,rounded corners=5pt,minimum width=3.6cm,minimum height=1.55cm},
  note/.style={draw,rounded corners=3pt,align=center,inner sep=4pt,text width=4.4cm},
  arr/.style={-{Latex[length=2mm]},thick},
  disk/.style={draw,thick}
]
  \node[obj] (U) at (0,0) {};
  \node[fill=white,inner sep=1pt] at ($(U.north)+(0,-.28)$) {\(U_{ijk}\)};
  \draw[disk] (-1.05,-.30) -- (-1.05,.38);
  \draw[disk] (0,-.30) -- (0,.38);
  \draw[disk] (1.05,-.30) -- (1.05,.38);
  \node[font=\scriptsize] at (-1.05,-.62) {\(D_i^v\)};
  \node[font=\scriptsize] at (0,-.62) {\(D_j^v\)};
  \node[font=\scriptsize] at (1.05,-.62) {\(D_k^v\)};
  \node[outputbox] (K) at (6.5,0) {\(K_v\cong D^3\)};
  \draw[arr] (U.east) -- (K.west);
  \node[align=center,font=\small] at (3.25,1.50)
    {cut along the complete\\three-disk row};
  \node[note,below=8mm of K] (relative)
    {chosen relative ball identification:\\\(E\) and its full bicollar are fixed};
  \draw[densely dashed] (K.south) -- (relative.north);
\end{tikzpicture}
\caption{The vertex-first cut-ball construction.  Every edge incident to
\(v\) uses this same complete row and the same relative ball identification.
The drawing records the cutting operation and fixed bicollar only; the disk
shapes are schematic.}
\label{fig:cut-ball-marking}
\end{figure}

\Needspace{4\baselineskip}
\begin{proposition}[Five complete vertex rows]
\label{prop:five-complete-rows}
The disks in each row of \eqref{tab:five-active-disk-rows} are embedded,
pairwise disjoint, oriented, and product framed.  The cut manifold \(K_v\)
is an oriented smooth \(3\)-ball.  Moreover, the ball identification may
be chosen to fix the global interface disk \(E\) and its full bicollar.
After adjoining the unchanged exterior row, each active triple therefore
gives a complete global row \(\mathfrak C_v\).
\end{proposition}

\paragraph{Band separation and relative ball gluing.}
The height and band separations first make the composite disks embedded and
product framed.  Cutting the central co-cores then reduces each row to the
same three elementary handles, while the prescribed cancellation order keeps
the terminal overlapping lobes distinct.  Because every move misses the
interface disk and its bicollar, the relative ball identifications glue to one
complete global row.

\begin{proof}
The height separations \eqref{eq:height-separation}, band separations
\eqref{eq:band-separation}, and disjoint smoothing boxes show directly
that every band sum in \eqref{eq:four-descendant-disks} is embedded.
The band orientations are the positive band-sum orientations, and the
simultaneous quadrant rounding makes the product normals on disk pieces
and bands agree.  Thus each composite disk has one global product-normal
collar.  This proves embeddedness, disjointness, orientation, and framing
without using homology.

Let
\begin{equation}
 C=\operatorname{Round}\bigl(\chi(Q\times[1-\tau,1])\bigr).
 \label{eq:central-three-ball}
\end{equation}
It is a ball, and \(U_{ijk}\) is obtained from it by attaching the three
product \(1\)-handles.  Cutting a handle along its middle co-core leaves
two half-handles, each a ball attached to the current ball along a disk;
absorbing the half-handle is a boundary-collar compression.  The row
\(v=0\) is therefore a complete co-core system.

For the four composite rows, after the carrier co-core has been cut, each
band lobe is boundary parallel in its supplied product box.  Sliding it
through the cleared half-handle retracts the composite disk to its root
while preserving the entire normal germ.  Applying this elementary
cancellation in the exact orders printed in
\cref{tab:five-cut-schedules} reduces every row to the three elementary
co-cores.  At the terminal vertex the \(k_\varepsilon\)-lobe of \(J\) is
nearer and is removed first; only then may the
\(k_{2\varepsilon}\)-lobe of \(I_\star\) pass through the cleared region.
No simultaneous cancellation or disjoint corridor for those two lobes
is assumed.  All isotopies are supported in the supplied handle, band,
and smoothing reserves, which miss \(E\), its bicollar, the old region,
and the unused handles.  This proves the relative ball assertion.

Finally cut the frozen exterior disks first.  The active remainder meets
the exterior cut product only along \(E\), and the five relative
identifications agree there.  Replacing one \(K_v\) by another therefore
does not alter the exterior cut product, proving completeness of the
global row.
\end{proof}

\subsection{Direct dual longitudes}

The dual curves must be constructed geometrically; their homology classes
alone do not supply embedded product-framed representatives.

\begin{lemma}[Direct dual-track update]
\label{lem:direct-dual-update}
Suppose a positive product-zero-framed slide replaces a moving disk
\(D_a\) by \(D_a'=D_a\#D_b^+\) and fixes the carrier \(D_b\).  If
\(\lambda_a,\lambda_b\) are disjoint product-framed duals before the
slide, then in a boundary-parallel track beside the meridian band one may
take
\begin{equation}
 \lambda_a'=\lambda_a,\qquad
 \lambda_b'=\lambda_b\natural(-\lambda_a)^+.
 \label{eq:direct-dual-update}
\end{equation}
After a product-germ-preserving bigon push, these are embedded disjoint
duals for \(D_a',D_b\).
\end{lemma}

\begin{proof}
Before the bigon push, \(D_a'\) meets \(\lambda_b'\) in two points.  The
carrier lobe \(D_b^+\) meets the \(\lambda_b\)-piece positively, while
the \(D_a\)-piece meets \((-\lambda_a)^+\) negatively.  Their joining
arcs bound the plumbing rectangle, whose interior misses all four
objects.  Push the longitude side across this bigon while keeping the
ambient normal coordinate fixed.  The two intersections cancel; the
positive diagonal intersections
\(D_a'\cdot\lambda_a'=D_b\cdot\lambda_b'=1\) remain, and all spectator
germs are unchanged.
\end{proof}

Choose the three joining rectangles in distinct reserved tracks and set
\begin{equation}
\begin{aligned}
 L_{ji}&=\ell_j\natural_{\alpha^\vee}(-\ell_i)^+,\\
 K_{ki}&=\ell_k\natural_{\beta_{ki}^\vee}(-\ell_i)^+,\\
 K_{kj}&=\ell_k\natural_{\beta_{kj}^\vee}(-\ell_j)^+.
\end{aligned}
\label{eq:three-direct-longitudes}
\end{equation}
Repeated application of \cref{lem:direct-dual-update}, still before any
edge, gives
\begin{equation}
\begin{array}{c|ccc}
v&\lambda_i^v&\lambda_j^v&\lambda_k^v\\
\hline
0&\ell_i&\ell_j&\ell_k\\
B&\ell_i&\ell_j&K_{kj}\\
A&\ell_i&L_{ji}&\ell_k\\
AC&\ell_i&L_{ji}&K_{ki}\\
\starv&\ell_i&L_{ji}&K_{kj}.
\end{array}
\label{tab:five-direct-dual-rows}
\end{equation}
The rectangles and their bigon boxes are disjoint, so the simultaneous
rows are embedded and product framed.  Put
\(m_s^v:=\partial D_s^v\) on the outgoing boundary.  More strongly,
\begin{equation}
 m_s^v\mathbin{\cdot}\lambda_r^v=\delta_{sr},\qquad
 m_s^v\cap\lambda_r^v=\varnothing\quad(s\neq r),
 \label{eq:geometric-dual-intersections}
\end{equation}
where the diagonal intersection is one positive transverse point in the
original handle chart.  The identities
\begin{equation}
 [L_{ji}]=[\ell_j]-[\ell_i],\quad
 [K_{ki}]=[\ell_k]-[\ell_i],\quad
 [K_{kj}]=[\ell_k]-[\ell_j]
 \label{eq:dual-homology-check}
\end{equation}
are checks on these already constructed curves, not substitutes for the
track construction.

After cutting, the diagonal intersection opens each \(\lambda_r^v\) into
an oriented arc
\begin{equation}
 c_{r,v}:D_{r,v}^-\longrightarrow D_{r,v}^+.
 \label{eq:cut-dual-ribbon-core}
\end{equation}
Its product neighborhood is the framed dual ribbon used in the
cut-sphere marking.

\subsection{The marked cut sphere}

Let \(S=S^2_{Q0}\) denote the abstract sphere cellulated in
\cref{app:cut-schedules}.  It has seven core faces
\begin{equation}
 E,\quad i^-,i^+,\quad j^-,j^+,\quad k^-,k^+,
 \label{eq:seven-cut-faces}
\end{equation}
three spoke--dual branches \(E\to r^-\to r^+\), one complementary
\(24\)-gon \(Z_{\mathrm{mod}}\), and cell count
\begin{equation}
 (V_{\mathrm{cell}},E_{\mathrm{cell}},F_{\mathrm{cell}})
 =(48,84,38).
 \label{eq:q0-cell-census}
\end{equation}

\Needspace{4\baselineskip}
\begin{proposition}[Five direct smooth realizations]
\label{prop:five-sphere-realizations}
For each \(v\in V\) there is an orientation-preserving diffeomorphism
\begin{equation}
 b_v:S\overset{\cong}{\longrightarrow}\partial K_v
 \label{eq:five-boundary-realizations}
\end{equation}
which sends every indexed face, ribbon, spoke, detector, and attachment
sector to its directly constructed counterpart on \(\partial K_v\).  The five maps
are chosen independently from the vertex rows and obey one common
signed-face convention.
\end{proposition}

\paragraph{The fat-tree disk and its complementary disk.}
The seven cut faces and three dual arcs first form a realized fat tree on each
cut sphere. The prescribed spoke slots make its regular neighbourhood a disk,
leaving one complementary disk; product charts identify the common ribbon
data, and an endpoint-flat boundary interpolation extends that identification
over the complement. The common detector and face convention are therefore
part of the smooth realization, not consequences of the cell count.

\begin{proof}
On \(\partial K_v\), begin with the seven realized cut faces and the
three disjoint dual arcs \eqref{eq:cut-dual-ribbon-core}.  They form four
forest components.  Choose one spoke
\(s_{r,v}:E\to D_{r,v}^-\) at a time, attaching on \(E\) in the cyclic
slots
\begin{equation}
 p_i=1,\qquad p_j=5,\qquad p_k=3.
 \label{eq:spoke-attachment-slots}
\end{equation}
Each spoke joins two distinct components in the connected complement,
so no cycle is created.  The resulting regular neighborhood of the
connected fat tree is a disk; its complement in the sphere
\(\partial K_v\) is a second disk \(Z_v\).  Thickening the six tree edges
gives the spoke and dual-ribbon rectangles.  The supplied \(E\)-collar
also carries one common detector with zero turning in both frame
directions.

Product-coordinate maps identify the regular neighborhood of the abstract
fat tree with the realized one and agree on complete overlap strips.  To
extend over the complementary disk, choose polar charts in which the
boundary germ has the form
\begin{equation}
 (r,[\theta])\longmapsto(r,[f_v(\theta)]),\qquad
 f_v(\theta+1)=f_v(\theta)+1,\quad f_v'(\theta)>0.
 \label{eq:complementary-disk-boundary-germ}
\end{equation}
With \(f_{v,s}=(1-s)\operatorname{id}+sf_v\) and an endpoint-flat radial
cutoff \(\eta\), use
\begin{equation}
 (r,[\theta])\longmapsto
 \bigl(r,[f_{v,\eta(r)}(\theta)]\bigr).
 \label{eq:positive-polar-fill}
\end{equation}
It is the identity near the polar center, hence smooth there, and for
\(r>0\) its angular derivative is
\((1-\eta(r))+\eta(r)f_v'(\theta)>0\).  It is therefore a positive disk
diffeomorphism agreeing to all orders with the boundary map on a full
annulus.  Gluing gives \(b_v\).  The explicit face list in
\cref{app:cut-schedules} proves the full incidence statement; a cell
count alone would not.
\end{proof}

For each \(r\), fix an orientation-reversing
\(\jmath_r:D^2\to D^2\) and positive face charts
\(\kappa_{r,v}^{\sigma,\partial}:D^2\to D_{r,v}^{\sigma}\) so that
\begin{equation}
 \kappa_{r,v}^{+,\partial}
 =\gamma_{r,v}\kappa_{r,v}^{-,\partial}\jmath_r^{-1},
 \label{eq:positive-face-chart}
\end{equation}
where \(\gamma_{r,v}\) is the canonical identification of the two cut
copies.  The surface pairings are
\begin{equation}
\begin{aligned}
 \pi_{r,v}^{-+,\partial}
  \kappa_{r,v}^{-,\partial}(z)
 &=\kappa_{r,v}^{+,\partial}(\jmath_rz),\\
 \pi_{r,v}^{+-,\partial}
 &=(\pi_{r,v}^{-+,\partial})^{-1}.
\end{aligned}
\label{eq:surface-face-pairings}
\end{equation}
They are smooth orientation-reversing surface maps.  They are not
cellular: a negative face has four sectors and a positive face has two.

The inherited two-sided face collars have a uniform positive width
\(d_{\mathrm{inh}}\).  In their local doubles define
\begin{equation}
 \pi_{r,v}^{-+,\mathrm{full}}
  \bigl(\kappa_{r,v}^{-,\partial}(z),u\bigr)
 =\bigl(\kappa_{r,v}^{+,\partial}(\jmath_rz),u\bigr),
 \qquad |u|<d_{\mathrm{inh}}.
 \label{eq:full-normal-pairing}
\end{equation}
The normal coordinate is preserved for both signs.  Negative \(u\)
belongs to the reflected double and is not claimed to be a point of
\(K_v\).

\subsection{One relative inward collar at each vertex}

\begin{lemma}[Relative collar extension]
\label{lem:vertex-relative-collar-extension}
Let \(M\) be compact with boundary, let
\(O_0\Subset O_1\subset\partial M\), and suppose that
\[
 c^{\mathrm{loc}}:O_1\times[0,a)\longrightarrow M
 \]
is a prescribed inward collar.  Then, for some \(b>0\), it extends to a
global inward collar
\[
 c:\partial M\times[0,b)\longrightarrow M
 \]
which equals \(c^{\mathrm{loc}}\) literally on
\(O_0\times[0,b)\).
\end{lemma}

\begin{proof}
Push \(\partial_u\) forward by \(c^{\mathrm{loc}}\) to a field
\(X_{\mathrm{loc}}\).  Choose
\(O_0\Subset O_{1/2}\Subset O_1\), a smaller depth \(a_0<a\), any
inward field \(Y\) near \(\partial M\), and a cutoff \(\rho\) equal to
one on
\(c^{\mathrm{loc}}(\overline{O_{1/2}}\times[0,a_0])\), with support in
the image of the local collar.  Extend \(\rho X_{\mathrm{loc}}\) by zero
and put
\begin{equation}
 X=\rho X_{\mathrm{loc}}+(1-\rho)Y.
 \label{eq:relative-collar-field}
\end{equation}
The inward half-space is convex, so \(X\) is strictly inward along the
boundary.  Compactness and the inverse function theorem give a common
short time for which its flow is a global collar.  On \(O_0\), uniqueness
of integral curves makes that flow equal to the prescribed local flow.
\end{proof}

The compatibility required for this application is ambient compatibility,
not a consequence of the boundary diffeomorphism \(b_v\).  Write each
inherited ambient product germ as
\[
 g_\alpha:B_\alpha\times[0,a_\alpha)\longrightarrow K_v.
\]
Every nonempty old--old overlap is a frozen handle, band, or smoothing
product box, and its transition fixes the inward variable to all orders.
The local cancellation isotopies and the direct-dual bigon pushes are also
the identity in that variable.  At a global \(E\)-attachment the detector
and initial spoke rectangle inherit the same inward variable from the fixed
\(E\)-bicollar.  The remaining auxiliary spoke and annular regions are
chosen only afterward and have no competing old ambient germ.  These cases
are exhaustive because the other full local boxes are disjoint.  Hence,
after one finite shrink, every overlap has the literal form
\begin{equation}
 g_\alpha(x,t)=g_\beta(h_{\beta\alpha}(x),t),
 \label{eq:vertex-compatible-product-germs}
\end{equation}
and the fields \((g_\alpha)_*(\partial_t)\) glue to one field
\(X_v^{\mathrm{fix}}\) on the union of the inherited product germs.

Choose positive-depth unions
\(
 \overline{G_v^{\mathrm{keep}}}\Subset G_v^{\mathrm{work}}
\)
inside those germs, with the keep set containing every overlap on which
literal equality is required.  On the newly chosen realized fat-tree and
protected-annulus pieces choose any smooth inward field \(Y_v\), and let
\(\chi_v^{\mathrm{keep}}\) be one on \(G_v^{\mathrm{keep}}\) and supported in
\(G_v^{\mathrm{work}}\).  Extending
\(\chi_v^{\mathrm{keep}}X_v^{\mathrm{fix}}\) by zero, set
\begin{equation}
 X_v^{\mathrm{pre}}
 =\chi_v^{\mathrm{keep}}X_v^{\mathrm{fix}}
  +(1-\chi_v^{\mathrm{keep}})Y_v.
 \label{eq:vertex-local-collar-field}
\end{equation}
The inward cone is convex, so a common short-time flow gives a local collar
equal to every old product germ on a smaller positive-depth neighborhood;
its restrictions define the previously unassigned auxiliary spoke and
annular germs.  The inward face-pairing domains and images use this same
cut-face collar.  Only the nonnegative halves of the two-sided pairings lie
in the realized cut ball; their negative halves remain in the corresponding
local doubles and are not asserted to be restrictions of an inward collar.
Thus all realized cut-face, \(E\)-interface, band, ribbon, spoke, detector,
pairing, annular, and complete marking germs are restrictions of this one
local collar, and the relative collar-extension lemma applies.

Taking the minimum over the five resulting global collars gives a common
width \(b_{\mathrm{col}}>0\).  Let \(d_{\mathrm{germ}}>0\) be the minimum
positive depth on which the local collars agree literally with the finitely
many inherited germs, decreased if necessary to lie below their common
inherited depth.  After restricting the inherited full-normal face collars
to one common positive depth, denote that source-backed pairing depth by
\(\eps_{\mathrm{pair}}^{\mathrm{src}}>0\).  Choose, before every edge,
\begin{equation}
\begin{gathered}
 0<c_P\ll a\ll h_0\ll h,\qquad
 3h<\min\{b_{\mathrm{col}},d_{\mathrm{germ}}\},\\
 0<\epsilon_c<\eps_{\mathrm{pair}}^{\mathrm{src}}<
 \delta_{\mathrm{rig}}<\epsilon_{\mathrm{work}}<c_P,
\end{gathered}
\label{eq:vertex-depth-hierarchy}
\end{equation}
and restrict the resulting collars to
\begin{equation}
 c_v:\partial K_v\times[0,3h]\longrightarrow K_v.
 \label{eq:five-global-inward-collars}
\end{equation}
On every protected germ \(c_v\) is the already prescribed product
collar, literally and to all orders.

In particular, define the actual inward face pairing throughout
\([0,c_P)\) by
\begin{equation}
\begin{aligned}
\Pi_{r,v}^{-+,(3)}
 \bigl(c_v(\kappa_{r,v}^{-,\partial}(z),u)\bigr)
 &=c_v(\kappa_{r,v}^{+,\partial}(\jmath_rz),u),\\
\Pi_{r,v}^{+-,(3)}&=(\Pi_{r,v}^{-+,(3)})^{-1}.
\end{aligned}
\label{eq:actual-inward-pairing}
\end{equation}
On \(0\leq u\leq\epsilon_c\), this is the inward restriction of
\eqref{eq:full-normal-pairing}.  Thus regluing uses the same inward
coordinate as every other protected marking.  The surface equation
\eqref{eq:surface-face-pairings} alone would not supply this ambient
compatibility.

\subsection{The pre-edge quotient and disk collar}

Choose a positive disk chart
\(\zeta_{\mathrm{mod}}:D^2\to Z_{\mathrm{mod}}\).  Using its polar radius,
choose nested protected regions
\begin{equation}
 \mathcal P_{\mathrm{mod}}\Subset\mathcal U_{\mathrm{mod}}
 \Subset\mathsf W\subset S,\qquad
 \iota_0(x)=(x,0),
 \label{eq:nested-vertex-domains}
\end{equation}
where \(\mathsf W\) omits a closed disk around the polar center and
contains the full protected annulus.  We use the direct specialization
\begin{equation}
 b_v^\partial=b_v,\qquad
 \xi_v=(b_v|_{Z_{\mathrm{mod}}})\zeta_{\mathrm{mod}},
 \qquad G_v=\id_{\R};
 \label{eq:direct-vertex-gauge}
\end{equation}
thus no endpoint gauge is solved from an incident edge.

Define the protected and thickened vertex embeddings by
\begin{align}
q_v^P(x,t)&=c_v(b_v(x),t),
 &&(x,t)\in\mathsf W\times[0,c_P),
 \label{eq:protected-vertex-embedding}\\
q_v^{(3)}(x,t)&=c_v(b_v^\partial(x),t),
 &&(x,t)\in\mathcal U_{\mathrm{mod}}\times[0,c_P),
 \label{eq:thickened-vertex-embedding}\\
q_v&=q_v^{(3)}\iota_0|_{\mathcal P_{\mathrm{mod}}}.
 \label{eq:boundary-vertex-embedding}
\end{align}
The two ambient maps agree on their common domain.

Reglue the three signed face pairs using
\eqref{eq:actual-inward-pairing}.  The resulting typed quotient is
\begin{equation}
\rho_v^{(3)}:K_v\longrightarrow
U_{ijk,v}:=
K_v/\langle\Pi_{i,v}^{(3)},\Pi_{j,v}^{(3)},\Pi_{k,v}^{(3)}\rangle,
\qquad
\rho_v=\rho_v^{(3)}|_{\partial K_v}.
\label{eq:pre-edge-three-dimensional-quotient}
\end{equation}
This is vertex data only; it asserts no compatibility with a future edge
map.

To define the outgoing parameterization, construct the same five marked
rows directly in a fixed abstract copy of the three-handle sector in
\(Y\).  Let
\(\widehat b_v:S\to\widehat S_v\) be its cut-sphere realization and
\(\widehat\rho_v:\widehat S_v\to Y\) its regluing quotient.  On the
exposed sector set
\begin{equation}
 p_v\bigl(\rho_vb_v(x)\bigr)
 =\widehat\rho_v\widehat b_v(x),\qquad x\in S,
 \label{eq:direct-outgoing-parameterization}
\end{equation}
and use the supplied common exterior parameterization elsewhere.  The
face-pairing conjugacies make
\eqref{eq:direct-outgoing-parameterization} independent of the chosen
representative of a quotient class.  Since they hold on open face
collars, the descended map is smooth; the two clauses agree to all orders
on the global \(E\)-bicollar.  Thus
\[
 p_v:\partial_+(W,\mathfrak C_v)\overset{\cong}{\longrightarrow}Y
\]
is one direct vertex parameterization.  In particular, \(p_\star\) is not
defined by either route.

Set
\begin{equation}
 \bar z_v
 =p_v\rho_vb_v\zeta_{\mathrm{mod}}:D^2\longrightarrow Y.
 \label{eq:outgoing-complementary-disk}
\end{equation}
Finally choose, once from the common vertex depth \(c_P\),
\begin{equation}
 u_0=\frac12c_P>0,\qquad
 z_v(x,t)=c_v(b_v\zeta_{\mathrm{mod}}(x),t),
 \quad (x,t)\in D^2\times[0,u_0].
 \label{eq:vertex-only-disk-collar}
\end{equation}
No later edge-dependent shrink changes \(u_0\), \(z_v\), or its domain.
At \(t=0\), the definitions give the correctly typed boundary identity
\begin{equation}
 \boxed{p_v\rho_vz_v(x,0)=\bar z_v(x).}
 \label{eq:corrected-vertex-boundary-crosswalk}
\end{equation}
The factor \(\rho_v\) cannot be omitted: \(z_v(x,0)\) lies on the cut
sphere, whereas \(p_v\) is defined after regluing.

Collect the cellulation, pairings, collars, and quotient data into
\(\cQ_v\).  These choices give one common set of endpoint data for the
later edge constructions.

\begin{proposition}[Simultaneous pre-edge vertex package]
\label{prop:simultaneous-vertex-package}
For the fixed tuple \(\cT\), there is one simultaneous choice
\begin{equation}
 \left(
 \mathfrak C_v,K_v,\cQ_v,q_v^P,q_v,q_v^{(3)},
 \rho_v^{(3)},\rho_v,p_v,\bar z_v,
 z_v:D^2\times[0,u_0]\to K_v
 \right)_{v\in V}
 \label{eq:constructed-vertex-package}
\end{equation}
with one common \(u_0>0\).  Every component is fixed before all edge
footprints, endpoint atlases, and edge representatives.
\end{proposition}

The parameterizations in this simultaneous construction identify the supplied
exterior data with one common abstract background marking
\(\mathfrak C_{\mathrm{bg}}\) on \(Y\).  It contains exactly the old,
unused-handle, and global-\(E\) data induced by
\(\mathfrak C_{\mathrm{ext}}\), and no varying active vertex row.  This is
the relative structure used when the two outgoing routes are compared in
\cref{sec:rigidity-pentagon}.

\subsection{The frozen exterior splitting}

The vertex construction also fixes, independently of all edges,
\begin{equation}
 \iota_{\mathrm{old}}:H_1(X;\Q)\hookrightarrow H_1(Y;\Q),\quad
 H_{\mathrm{old}}=\im\iota_{\mathrm{old}},\quad
 \lambda_g^Y=\iota_{\mathrm{old}}(\lambda_g).
 \label{eq:old-homology-inclusion}
\end{equation}
For
\(\mathcal I_{\mathrm{unused}}
=\{1,\ldots,N\}\setminus\{i,j,k\}\), put
\begin{equation}
\begin{aligned}
H_{\mathrm{ext}}
 &=H_{\mathrm{old}}\oplus
 \bigoplus_{t\in\mathcal I_{\mathrm{unused}}}
   \bigl(\Q[m_t]\oplus\Q[\ell_t]\bigr),\\
I_{\mathrm{ext}}&=\id_{H_{\mathrm{ext}}},\\
\Lambda_{\mathrm{ext}}
 &=\lambda_g^Y\oplus
 \bigoplus_{t\in\mathcal I_{\mathrm{unused}}}\Q[m_t].
\end{aligned}
\label{eq:frozen-exterior-splitting}
\end{equation}
Writing \(m_r,\lambda_r\) for the \(p_0\)-images of the active framed row,
set
\begin{equation}
 U=\Span_{\Q}\{[m_i],[m_j],[m_k]\},\qquad
 U^\vee=\Span_{\Q}\{[\lambda_i],[\lambda_j],[\lambda_k]\}.
 \label{eq:active-homology-summands}
\end{equation}
Then
\begin{equation}
 H_1(Y;\Q)=H_{\mathrm{ext}}\oplus(U\oplus U^\vee),
 \qquad
 \Lambda=\Lambda_{\mathrm{ext}}\oplus U.
 \label{eq:pre-edge-homology-splitting}
\end{equation}
When \(N=3\), the unused sums in
\eqref{eq:frozen-exterior-splitting} are empty, and
\(\dim\Lambda=g+3=g+N\).  The later edge actions are compared in this
fixed homology splitting; preservation of \(\Lambda\) is proved in
\cref{sec:homology}.