\section{Finite compatibility conditions for the three map stages}
\label{sec:pc5}

The five edge constructions must agree with all protected charts before
their route composites can be compared.  We collect these agreements for
the same cut, quotient, and global maps already constructed; no additional
edge representative is chosen.

We first restrict the cut-level chart identities to depths common to their
respective categories.  The nine families below distinguish faces and ribbons,
continuations and spokes, detector and carrier data, pairings, and global
identities.  This distinction keeps each equation on the domain of the map
that satisfies it, especially at the exterior attachment.

\subsection{Common subordinate depths}

Because there are only five edges and finitely many charts in each marked
cut ball, the positive chart widths have positive minima.  Choose
\(\eps_{\mathrm{face}},\eps_{\mathrm{rib}},\eps_\tau,
\eps_{\mathrm{spoke}},\eps_{\mathrm{det}}>0\) so that
\begin{equation}
 \eps_{\mathrm{face}}\leq u_{\mathrm{pair}},
 \qquad
 \max\{\eps_{\mathrm{rib}},\eps_\tau,
        \eps_{\mathrm{spoke}},\eps_{\mathrm{det}}\}
 \leq u_{\mathrm{cmp}},
 \label{eq:pc5-category-depths}
\end{equation}
and each is strictly smaller than the minimum width of the charts in its
own category.  Independently set
\begin{equation}
 \eps_{\mathrm{PC5,work}}=\frac12u_{\mathrm{cmp}},
 \qquad
 \eps_{\mathrm{PC5,pair}}=\frac12u_{\mathrm{pair}}.
 \label{eq:pc5-fresh-depths}
\end{equation}
Thus G6 lies strictly inside the comparison depth and G7 lies strictly
inside the source-backed pairing depth.  Neither number is a renamed
source work width.

For compactness, retain the notation
\(\mathscr L_e\) and \(\mathscr S_e\) of
\eqref{eq:local-spectator-maps}.  On a signed face put
\begin{equation}
 \mathscr O_{e,r}=
 \begin{cases}
  \mathscr L_e,&r=p_e,q_e,\\
  \mathscr S_e,&r=s_e.
 \end{cases}
 \label{eq:face-owner}
\end{equation}

\subsection{The nine compatibility families}

\begin{proposition}[Finite compatibility scheme]
\label{prop:pc5-scheme}
For every edge \(e:v\to w\), the following nine families hold on the
displayed complete product domains, with equality of all jets.

\begin{enumerate}[label=\textup{G\arabic*},leftmargin=3em]
\item \emph{Faces, seven records.}
For \(r=p_e,q_e,s_e\) and \(\sigma\in\{-,+\}\),
\begin{equation}
 \mathscr O_{e,r}\kappa_{r,v}^{\sigma,(3)}(z,u)
 =\kappa_{r,w}^{\sigma,(3)}(z,u),
 \quad 0\leq u\leq\eps_{\mathrm{face}}.
 \label{eq:pc5-g1-faces}
\end{equation}
The seventh record is the global-\(E\) cut equation
\eqref{eq:global-E-cut} on its complete cut half-germ.

\item \emph{Ribbons, six records.}
For \(a\in\{s_i,c_i,s_j,c_j,s_k,c_k\}\),
\begin{equation}
 \Fcut e\rho_{a,e,v}^{(3)}(x,t,u)
 =\rho_{a,e,w}^{(3)}(x,t,u),
 \qquad 0\leq u\leq\eps_{\mathrm{rib}},
 \label{eq:pc5-g2-ribbons}
\end{equation}
and the differential fixes the ordered frame
\((\partial_x,\partial_t,\partial_u)\).  Active carrier ribbons are
local-owned, active spoke ribbons use the two branches and are
split-owned, and the two spectator ribbons are spectator-owned.
In the terminal \(AC\star\) carrier row, the target chart contains the
fixed product-normal exceptional-ribbon comparison.

\item \emph{Active continuations, two records.}
For \(r=p_e,q_e\),
\begin{equation}
 (\Xi^z_{e,v})^{-1}\tau_{e,r,v}
 =\tau_{e,r}^{\mathrm{sp}}
 =(\Xi^z_{e,w})^{-1}\tau_{e,r,w},
 \qquad
 \mathscr S_e\tau_{e,r,v}=\tau_{e,r,w}.
 \label{eq:pc5-g3-continuations}
\end{equation}
This is equality in common spectator coordinates, not equality of arcs in
different realized balls.  It holds on the complete continuation
product germ through depth \(\eps_\tau\), with its full frame and all jets.

\item \emph{Spokes and the spectator word, twelve records.}
For each active \(r=p_e,q_e\) and \(a\in\{v,w\}\),
\begin{equation}
 s_{r,a}=\tau_{e,r,a}*s_{e,r,a}^{\mathrm{loc}},
 \qquad
 \Fcut e(s_{r,v})=s_{r,w}.
 \label{eq:pc5-g4-active-spokes}
\end{equation}
The two continuations are spectator-owned and the two local segments are
local-owned.  The spectator spoke, the complete spectator branch, and its
eight ordered letter germs are spectator-owned and are carried
productwise with their positions unchanged.  Thus the count is
\(2+1+1+8=12\), not twelve further records in addition to the eight
letters.  Every one of these statements is on its full product germ
through depth \(\eps_{\mathrm{spoke}}\).

\item \emph{Detector, one record.}
On the complete detector chart,
\begin{equation}
 \mathscr S_e\delta_v(\theta,n,u)=\delta_w(\theta,n,u),
 \qquad
 d\mathscr S_e(\partial_\theta,\partial_n,\partial_u)
 =(\partial_\theta,\partial_n,\partial_u),
 \label{eq:pc5-g5-detector}
\end{equation}
the core is transported, and the turn remains zero.
The displayed chart includes the closed parameter face
\(0\leq u\leq\eps_{\mathrm{det}}\).

\item \emph{Full protected carrier, one record.}
Let \(\mathcal A_{\mathrm{mod}}^{\mathrm{open}}\) be the fixed nonempty
relative-open neighborhood of the entire closed protected annulus.  Then
\begin{equation}
 \Fcut e q_v^P(\zeta_{\mathrm{mod}}(x),u)
 =q_w^P(\zeta_{\mathrm{mod}}(x),u),
 \quad
 x\in\mathcal A_{\mathrm{mod}}^{\mathrm{open}},\quad
 0\leq u\leq\eps_{\mathrm{PC5,work}}.
 \label{eq:pc5-g6-carrier}
\end{equation}
This record is split-owned: its local and spectator restrictions agree on
a nonempty open overlap to every order.

\item \emph{Pairings, three records.}
For \(r=i,j,k\),
\begin{align}
 \Fcut e^{\,r,+}\Pi_{r,v}^{-+,(3)}
     \kappa_{r,v}^{-,(3)}(z,u)
 &=\Pi_{r,w}^{-+,(3)}\Fcut e^{\,r,-}
     \kappa_{r,v}^{-,(3)}(z,u)\notag\\
 &=\kappa_{r,w}^{+,(3)}(\jmath_rz,u),
 \quad 0\leq u\leq\eps_{\mathrm{PC5,pair}}.
 \label{eq:pc5-g7-pairings}
\end{align}
The normal remains \(u\mapsto u\).  The two active records are
local-owned and the spectator record is spectator-owned.

\item \emph{Stage-typed global data, four records.}
The global-\(E\) cut trace is owned by \(\Fcut e\), through
\(\mathscr S_e\).  Only the global map owns the remaining three records:
\begin{equation}
 \Fglob e|_{\cN^{\mathrm{two}}(E)}=\id,
 \qquad
 \Fglob e|_{W_{\mathrm{old}}}=\id,
 \qquad
 \Fglob e|_{\bigcup_{t\notin\{i,j,k\}}\cN_t}=\id.
 \label{eq:pc5-g8-global}
\end{equation}
No \(\Xi^z\)- or \(\Fcut e\)-formula is asserted on the old or unused
regions.

\item \emph{Typed role transport, one record.}
On the complete named-carrier product germ, the edge map carries the
source active curve to the prescribed target curve, fixes its indicated
spectator, and preserves the carrier and product normal.  The five rows
are
\begin{center}
\begin{tabularx}{\linewidth}{@{}l@{\quad}X@{}}
\(0B\)&\(j_0\to J\) over \(k_0,k_\eps\) via \(\beta_0\);
        spectator \(i_0\).\\
\(B\star\)&\(i_0\to I_\star\) over complete \(J,J^+\) via \(\alpha\),
        where \(J^+=j_\eps\#_{\beta_1}k_{2\eps}\);
        \(\alpha\) ends on the \(j_\eps\) lobe; spectator \(k_0\).\\
\(0A\)&\(i_0\to I_A\) over \(j_0,j_\eps\) via \(\alpha\);
        spectator \(k_0\).\\
\(AAC\)&The whole \(I_A\), including \(\Omega_j\) and its
        \(j_\eps\) lobe and marking, maps to \(I_\star\) over
        \(k_0,k_{2\eps}\) via \(\beta_1\); spectator \(j_0\).\\
\(AC\star\)&\(j_0\to J\) over \(k_0,k_\eps\) via \(\beta_0\);
        actual spectator \(I_\star\).\\
\end{tabularx}
\end{center}
where the last row includes the product-normal exceptional-ribbon comparison
\(\widetilde K\to K_{kj}\), with \(\widetilde K\) as in
\eqref{eq:raw-sharp-longitude}, and leaves the pre-edge
\(q_\star,p_\star\) unchanged.  These records are owned by
the edge construction itself; they do not enlarge the representative's
support beyond the named carrier germ.
\end{enumerate}
\end{proposition}

\begin{proof}
G1 is the union of the six signed-face consequences used in
\cref{prop:pair15} and the cut equation \eqref{eq:global-E-cut}.  G2--G5
are the product-germ statements proved in
\cref{prop:cut-stabilization}; the local/spectator decomposition of the
active spokes gives precisely the asserted split ownership.  G6 is
\eqref{eq:pc3-comparison} restricted to the fixed full-carrier germ and
to the strictly smaller depth in \eqref{eq:pc5-fresh-depths};
\eqref{eq:z-full-disk-comparison} supplies its closed disk-collar
extension.  G7 is \cref{prop:pair15} restricted to the subordinate pairing
depth.  G8 is \cref{prop:global-extension} together with the separately
proved cut trace.  The named-carrier construction and the exceptional-ribbon
lemma give the five G9 rows, including the product normal in the terminal
row.  These derivations also establish the owners displayed above.  In
particular, no family chooses a new map.  The literal shared-incidence
equalities needed when two successive edges meet are
\eqref{eq:stabilization-incidence-aliases}; they identify the frozen
vertex charts, never the genuinely edge-dependent continuation pieces.
\end{proof}

The per-edge sizes are
\[
 (7,6,2,12,1,1,3,4,1),
\]
so five edges give \(45\) group headers and
\begin{equation}
 5(7+6+2+12+1+1+3+4+1)=185
 \label{eq:pc5-total-count}
\end{equation}
typed records.  When \(N=3\), the five unused-handle G8 records
remain included in this finite compatibility enumeration, but their domains
are empty; the other
records retain their stated nonempty product domains.
% Reader-reference migration: \cref{app:pc5-census} remains an internal alias
% for the repository-retained finite table omitted from the public driver.

\subsection{Shared vertex data and the radial collar}

Set, in right-factor-first order,
\begin{equation}
 \widetilde F_L=\widetilde F_{B\star}\compose\widetilde F_{0B},
 \qquad
 \widetilde F_R=\widetilde F_{AC\star}\compose
                    \widetilde F_{AAC}\compose\widetilde F_{0A},
 \qquad
 \cQ_{\mathrm{cut}}=\widetilde F_R^{-1}\compose\widetilde F_L.
 \label{eq:cut-route-discrepancy}
\end{equation}

\begin{corollary}[Cut-owned radial compatibility]
\label{cor:radial-collar}
The two cut routes in \eqref{eq:cut-route-discrepancy} use the literal
same vertex package at every common endpoint.  On every protected chart
listed in G1--G7 and on the cut trace in G8, their coordinate expressions
agree.  Hence \(\cQ_{\mathrm{cut}}\) is the identity on those complete
product germs.
\end{corollary}

\begin{proof}
At an intermediate vertex, the outgoing chart of one edge and the incoming
chart of the next are restrictions of the same previously chosen vertex
chart.  Thus composing the edge equations telescopes: both routes carry
each protected coordinate at vertex \(0\) to the same coordinate at
vertex \(\star\).  Applying \(\widetilde F_R^{-1}\) gives the identity.
For the radial variables, the transition reparameterizations are the
literal identity, and G6 supplies a full open annular germ rather than only
its boundary circle.  The all-jet overlap statements in
\cref{prop:pc5-scheme} make these identities compatible on chart
intersections.  The conclusion is cut-owned; G8's old and unused rows are
not subsets of \(K_0\) and are not part of it.
\end{proof}