\section{Edge maps, full-normal pairings, and stabilization} \label{sec:edge-pairing-stabilization} For each edge, the local and spectator maps have the same product formula on the internal bicollar, so they glue to a smooth cut-ball map. To reglue the three cut disks, this map must also intertwine their pairings on full normal collars. The five edge constructions give the fifteen pairing conjugacies needed for smooth quotient descent. Fix a directed edge \(e:v\to w\) of \(E_5\). All vertex data and all endpoint chart-and-marking systems used below have already been chosen in the order of \eqref{eq:witness-order-main}. In particular, no construction in this section changes \(u_0\), \(z_v\), \(z_w\), or either endpoint marking. \subsection{The amalgamated cut map} For each endpoint \(a\in\{v,w\}\), the cut ball has the decomposition \begin{equation} K_a=K_{e,a}^{\mathrm{loc}}\mathop{\cup}_{E_{e,a}} L_{e,a}^{\mathrm{sp}}, \qquad E_{e,a}\neq E. \label{eq:cut-ball-two-halves} \end{equation} Here \(E_{e,a}\) is the internal local--spectator interface, whereas \(E\) is the global exterior attachment disk. The two closed halves in \eqref{eq:cut-ball-two-halves} meet precisely along \(E_{e,a}\). The relative constructions of \cref{sec:naturalization,sec:relative-extensions} also give a signed open bicollar \[ c_{e,a}^{E_e}:E_e\times(-\eta_e,\eta_e)\longrightarrow K_a \] to which both half-maps extend. Define the local and spectator maps \begin{equation} \mathscr L_e= \widehat\Theta_{e,w}\widehat\phi_e^{\mathrm{cut}} \widehat\Theta_{e,v}^{-1}, \qquad \mathscr S_e=\Xi^z_{e,w}(\Xi^z_{e,v})^{-1}. \label{eq:local-spectator-maps} \end{equation} The hats in the first formula are essential: the representative \(\widehat\phi_e^{\mathrm{cut}}\) was selected only after all ten endpoint chart-and-marking systems had been frozen. \begin{proposition}[Cut-map gluing] \label{prop:cut-map-gluing} The formula \begin{equation} \Fcut e(x)= \begin{cases} \mathscr L_e(x),&x\in K_{e,v}^{\mathrm{loc}},\\ \mathscr S_e(x),&x\in L_{e,v}^{\mathrm{sp}} \end{cases} \label{eq:cut-map-piecewise} \end{equation} defines an orientation-preserving diffeomorphism \(K_v\to K_w\). On the complete signed interface bicollar, \begin{equation} \mathscr L_e c_{e,v}^{E_e}(z,t) =c_{e,w}^{E_e}(z,t) =\mathscr S_e c_{e,v}^{E_e}(z,t), \qquad |t|<\eta_e, \label{eq:interface-all-jets} \end{equation} and hence the two branches agree to every order. In spectator coordinates \((\Xi^z_{e,w})^{-1}\mathscr S_e\Xi^z_{e,v}=\id\). \end{proposition} \begin{proof} The local and spectator maps have different geometric roles, but their extensions use the same signed coordinate on the internal bicollar. We first check that these two formulas are one smooth map on that open overlap; the same check for their inverses then proves that their union is a diffeomorphism. Edgewise naturality, transported through the endpoint-specific local charts \(\widehat\Theta_{e,v}\) and \(\widehat\Theta_{e,w}\), gives the first equality in \eqref{eq:interface-all-jets}. The relative decorated-ball extension was fixed on the entire signed bicollar, so its spectator atlas gives the second equality, with the same variables \((z,t)\). Thus, on an open neighborhood of the common disk, both branches of \eqref{eq:cut-map-piecewise} are the single coordinate map \(c_{e,v}^{E_e}(z,t)\mapsto c_{e,w}^{E_e}(z,t)\). The piecewise formula is therefore smooth, not merely continuous across \(E_e\), and all of its jets match there. Each branch is orientation preserving on its half. Since they use the same signed normal coordinate on the overlap, their orientations agree. The corresponding formula with \(v,w\) exchanged and each constituent inverted is a smooth two-sided inverse. This proves the claim. The spectator-coordinate identity follows immediately from the definition of \(\mathscr S_e\). \end{proof} The spectator half in \eqref{eq:cut-ball-two-halves} contains the global \(E\)-cut attachment and the protected spectator data. It does not contain the quotient exterior, the old region, or any unused handle. Consequently \cref{prop:cut-map-gluing} makes no assertion about those global regions. \subsection{Inner and outer collar comparison} Let \(C_+^D\subset D^2\) be the buffered inner disk, let \(\mathcal W_-\) be the outer work region on the protected boundary, and let \(\mathcal A_{-+}^S\) be their open surface overlap. The relative-band construction gives the two covers \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:inner-outer-covers} \end{equation} \begin{lemma}[Full-disk comparison] \label{lem:full-disk-comparison} For every \(e:v\to w\), \begin{align} \Fcut e q_v^P&=q_w^P &&\text{on }\mathsf W\times[0,u_{\mathrm{cmp}}), \label{eq:pc3-comparison}\\ \Fcut e z_v&=z_w &&\text{on }D^2\times[0,u_{\mathrm{cmp}}]. \label{eq:z-full-disk-comparison} \end{align} \end{lemma} \begin{proof} The endpoint core equations control the inner region, and the local and restricted spectator equations control the outer region. The two covers above join these comparisons into one formula on the full disk; continuity then includes its closed depth face. On \(C_+^S\cap\mathsf W\), the inner-core equation for the two endpoint chart-and-marking systems and the spectator branch of \eqref{eq:cut-map-piecewise} give \(\Fcut e q_v^P=q_w^P\). On \(\mathcal W_-\), the local formula and the restricted outer spectator formula give the same equality. The two calculations agree on the open seam \(\mathcal A_{-+}^S\) by \eqref{eq:interface-all-jets}; the first cover in \eqref{eq:inner-outer-covers} proves \eqref{eq:pc3-comparison}. On \(C_+^D\), the two endpoint inner-core equations give directly \[ \Fcut e z_v =\Xi^z_{e,w}(\Xi^z_{e,v})^{-1}z_v =\Xi^z_{e,w}\iota_e^{\mathrm{core}} =z_w. \] On the second member of the disk cover, \(\zeta_{\mathrm{mod}}(x)\in\mathsf W\), where the vertex construction does identify \(q_a^P(\zeta_{\mathrm{mod}}(x),t)=z_a(x,t)\); there \eqref{eq:z-full-disk-comparison} follows from \eqref{eq:pc3-comparison}. The two formulas agree on their open overlap. They hold for \(0\leq t0\) fixed in \eqref{eq:vertex-depth-hierarchy}, and put \begin{equation} u_{\mathrm{pair}} =\frac12\min\{u_{\mathrm{cmp}}, \eps_{\mathrm{pair}}^{\mathrm{src}}\}>0. \label{eq:pair-depth-section9} \end{equation} For \(r\in\{i,j,k\}\) and a vertex \(a\), let \(\kappa_{r,a}^{\sigma,(3)}\), \(\sigma\in\{-,+\}\), be the positive inward half-collar charts. The supplied pairing and its inverse satisfy \begin{equation} \Pi_{r,a}^{-+,(3)}\kappa_{r,a}^{-,(3)}(z,t) =\kappa_{r,a}^{+,(3)}(\jmath_rz,t), \qquad \Pi_{r,a}^{+-,(3)}=(\Pi_{r,a}^{-+,(3)})^{-1}, \label{eq:source-full-normal-pairing} \end{equation} for \(0\leq t\leq u_{\mathrm{pair}}\). In particular, the normal coordinate is \(t\mapsto t\), not \(t\mapsto-t\). Write \[ \Fcut e^{\,r,\sigma} :=\Fcut e\big|_{\im\kappa_{r,v}^{\sigma,(3)}} \] for the corresponding signed-face restriction. The source label \(v\) in this definition is determined by the directed edge \(e:v\to w\). \begin{proposition}[Full-normal pairing conjugacy] \label{prop:pair15} For every edge \(e:v\to w\) and every \(r\in\{p_e,q_e,s_e\}\), both directed identities \begin{align} \Fcut e^{\,r,+}\Pi_{r,v}^{-+,(3)} &=\Pi_{r,w}^{-+,(3)}\Fcut e^{\,r,-}, \label{eq:pair-forward}\\ \Fcut e^{\,r,-}\Pi_{r,v}^{+-,(3)} &=\Pi_{r,w}^{+-,(3)}\Fcut e^{\,r,+} \label{eq:pair-backward} \end{align} hold on the complete half-collar of depth \(u_{\mathrm{pair}}\), with all jets. Thus there are exactly \(5\cdot3=15\) conjugacy pairs. \end{proposition} \begin{proof} For \(r=p_e,q_e\), edgewise naturality gives \(\Fcut e\kappa_{r,v}^{\sigma,(3)} =\kappa_{r,w}^{\sigma,(3)}\). For \(r=s_e\), the same formula follows from the selected spectator atlas. Hence \begin{align*} \Fcut e^{\,r,+}\Pi_{r,v}^{-+,(3)} \kappa_{r,v}^{-,(3)}(z,t) &=\Fcut e\kappa_{r,v}^{+,(3)}(\jmath_rz,t)\\ &=\kappa_{r,w}^{+,(3)}(\jmath_rz,t)\\ &=\Pi_{r,w}^{-+,(3)}\kappa_{r,w}^{-,(3)}(z,t)\\ &=\Pi_{r,w}^{-+,(3)}\Fcut e^{\,r,-} \kappa_{r,v}^{-,(3)}(z,t). \end{align*} The chart is onto its half-collar image, proving \eqref{eq:pair-forward}. Composing with the two inverse pairing maps gives \eqref{eq:pair-backward}. Every displayed map is productwise in \(t\), so differentiating in the common product coordinates proves equality of all jets. The role table \eqref{eq:underlying-role-triples} has three labels in each of five rows, giving the asserted count. \end{proof} The common spectator chart also supplies a full product chart \(\epsilon_{E,e,a}^{\mathrm{cut}} =\Xi^z_{e,a}\epsilon_E^{\mathrm{sp}}\) for the global-\(E\) cut attachment. The spectator-coordinate identity in \cref{prop:cut-map-gluing} gives \begin{equation} \Fcut e\epsilon_{E,e,v}^{\mathrm{cut}} =\epsilon_{E,e,w}^{\mathrm{cut}} \label{eq:global-E-cut} \end{equation} on its complete product domain, with every derivative. This remains a cut-attachment statement; it is not yet an exterior identity. \subsection{Stabilization before descent} For an active label \(r\in\{p_e,q_e\}\), the local interface word begins at \(E_e\), \begin{equation} \mathcal B_{e,r}^{\mathrm{loc}} =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:local-branch-word} \end{equation} For every \(r\in\{i,j,k\}\), the global branch word instead begins at \(E\), \begin{equation} \mathcal B_r^{\mathrm{glob}} =f_{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:global-branch-word} \end{equation} These words have the same formal length but different initial objects. Their five amalgamations, including the position of the spectator word, are \begin{equation} \begin{array}{c|c|c|c} e&\text{two local words}&\text{spectator insertion} &\text{resulting global order}\\ \hline 0B&\mathcal B_{e,j}^{\mathrm{loc}}\mathcal B_{e,k}^{\mathrm{loc}} &\mathcal B_i^{\mathrm{glob}}\ \text{before} &\mathcal B_i^{\mathrm{glob}}\mathcal B_j^{\mathrm{glob}} \mathcal B_k^{\mathrm{glob}}\\ B\star&\mathcal B_{e,i}^{\mathrm{loc}}\mathcal B_{e,j}^{\mathrm{loc}} &\mathcal B_k^{\mathrm{glob}}\ \text{after} &\mathcal B_i^{\mathrm{glob}}\mathcal B_j^{\mathrm{glob}} \mathcal B_k^{\mathrm{glob}}\\ 0A&\mathcal B_{e,i}^{\mathrm{loc}}\mathcal B_{e,j}^{\mathrm{loc}} &\mathcal B_k^{\mathrm{glob}}\ \text{after} &\mathcal B_i^{\mathrm{glob}}\mathcal B_j^{\mathrm{glob}} \mathcal B_k^{\mathrm{glob}}\\ AAC&\mathcal B_{e,i}^{\mathrm{loc}}\mathcal B_{e,k}^{\mathrm{loc}} &\mathcal B_j^{\mathrm{glob}}\ \text{between} &\mathcal B_i^{\mathrm{glob}}\mathcal B_j^{\mathrm{glob}} \mathcal B_k^{\mathrm{glob}}\\ AC\star&\mathcal B_{e,j}^{\mathrm{loc}}\mathcal B_{e,k}^{\mathrm{loc}} &\mathcal B_i^{\mathrm{glob}}\ \text{before} &\mathcal B_i^{\mathrm{glob}}\mathcal B_j^{\mathrm{glob}} \mathcal B_k^{\mathrm{glob}}. \end{array} \label{eq:five-word-amalgamations} \end{equation} \begin{proposition}[Cut-level stabilization] \label{prop:cut-stabilization} The maps \(\Fcut e\) preserve the complete cut-level marking: the two active local words, the inserted spectator word, all six framed ribbons, the complete spokes, the zero-turn detector, all forty spectator letter germs, the global-\(E\) cut chart, and the fifteen pairing conjugacies. The local rotation census is \((32,56,26)\); inserting the spectator annular graph contributes \((16,28,12)\), giving \begin{equation} (32,56,26)+(16,28,12)=(48,84,38). \label{eq:stabilization-census} \end{equation} No old-region or unused-handle assertion occurs at this stage. \end{proposition} \begin{proof} The common signed interface model identifies the cyclic orders on the two halves with opposite induced orientations. Gluing those cyclic orders, not merely adding Euler characteristics, inserts the rank-one spectator annular graph into the two-active-branch tree. The explicit core, collar, ribbon, and complementary cycles in \cref{eq:appendix-core-face-cycles,eq:appendix-collar-face-cycles,% eq:appendix-ribbon-face-cycles,eq:appendix-complementary-24-cycle} enumerate the inserted vertices, edges, and faces as \((16,28,12)\); \cref{prop:appendix-finite-sphere} verifies that these cycles exhaust the rounded boundary. This proves \eqref{eq:stabilization-census} and the prescribed twenty-four-letter global rotation word. The relative positions in \eqref{eq:five-word-amalgamations} are exactly the positions at which the spectator cycles enter that enumeration. For every active incidence, the endpoint charts split the already fixed complete realized spoke uniquely as \[ s_{r,a}=\tau_{e,r,a}*s_{e,r,a}^{\mathrm{loc}}. \] The spectator coordinate identity transports \(\tau_{e,r,v}\) to \(\tau_{e,r,w}\), while the local branch transports \(s_{e,r,v}^{\mathrm{loc}}\) to \(s_{e,r,w}^{\mathrm{loc}}\). Their concatenation therefore transports the complete spoke. The same product-coordinate calculation transports the six ribbon charts, the detector, and each of the eight ordered germs in the spectator word. It also preserves the insertion position of every letter. Repeated incidences merely give different coordinate decompositions of the same vertexwise spoke; they do not reselect it or identify the genuinely edge-dependent pieces. Finally, \cref{prop:pair15} and \eqref{eq:global-E-cut} supply the pairing and global-\(E\) cut clauses. All these equalities hold on full product germs, so all derivatives agree. For completeness, the incidence-independence just used is literal. Let \(\mathcal R_6=\{s_i,c_i,s_j,c_j,s_k,c_k\}\), and let \(\mathcal L_r\) be the eight ordered letters of \(\mathcal B_r^{\mathrm{glob}}\). If \(e,e'\) meet at \(a\), then on their common product domains the middle symbols in the following display \emph{define} the indicated common vertexwise values: \begin{align} \rho_{\alpha,e,a}^{(3)} &=\rho_{\alpha,a}^{(3)} =\rho_{\alpha,e',a}^{(3)} &&(\alpha\in\mathcal R_6),\notag\\ \epsilon_{E,e,a}^{\mathrm{cut}} &=\epsilon_{E,a}^{\mathrm{cut}} =\epsilon_{E,e',a}^{\mathrm{cut}},\notag\\ \delta_{e,a}&=\delta_a=\delta_{e',a}, &b_{\delta_E,e,a}&=b_{\delta_E,a}=b_{\delta_E,e',a},\notag\\ \chi_{\ell,e,a}^{\mathrm{sp}} &=\chi_{\ell,r,a}^{\mathrm{sp}} =\chi_{\ell,e',a}^{\mathrm{sp}} &&(\ell\in\mathcal L_r),\notag\\ \tau_{e,r,a}*s_{e,r,a}^{\mathrm{loc}} &=s_{r,a} =\tau_{e',r,a}*s_{e',r,a}^{\mathrm{loc}} &&\text{for a repeated }(r,a). \label{eq:stabilization-incidence-aliases} \end{align} The face, pairing, and protected-collar charts already carry only the vertex label, so no further cross-edge identification is required. Equation \eqref{eq:stabilization-incidence-aliases} identifies the fixed vertexwise objects, not the edge-dependent continuations or local pieces. The assertion concerns only the two halves of the cut ball and their markings. The old region and unused handles lie outside that ball and therefore cannot enter the conclusion. \end{proof}