\section{The local positive two-site slide} \label{app:local-positive-slide} This appendix proves the local input stated in \cref{thm:positive-two-site-input}. The point of the construction is to specify a relative diffeomorphism from geometric data, rather than to infer one from the four homology classes in \eqref{eq:local-positive-slide-rows}. We therefore keep separate the small ball which constructs the slid disk, the larger two-handle sector which supports the map, and the two balls obtained by cutting that sector. Write \(W_{\mathfrak C}\) for the frozen oriented compression-body sector with the old complete marking, and \(W_{\mathfrak C^\gamma}\) for the same sector with the slid complete marking. Fix throughout the exact push-off, band, smoothing boxes, dual band, and exterior-interface germ supplied in \cref{thm:positive-two-site-input}. Every coordinate chosen below is adapted to those objects. In particular, no occurrence of \(D_2^{\mathrm{po}},\gamma,\gamma^\vee\), or \(E_\gamma\) denotes a newly selected replacement. The construction has five geometric transitions. First the prescribed band sum produces the slid disk inside a small regular neighborhood. Next the old and slid disk systems are shown to cut the same product sector, while a larger sector supplies the two cut balls on which the map will live. The finite boundary dictionary then records the actual faces, ribbons, pairings, and detector collar. Finally, the cellular boundary map is smoothed relative to those data, extended across the cut balls, and reglued. Keeping these roles separate is essential: the small neighborhood constructs the disk, but the larger cut-ball extension is what supplies the relative diffeomorphism. \subsection{The product model and the small slide ball} Let \(Q\) be a coordinate disk in the high boundary which contains the four supplied handle feet and the prescribed joining-band neighborhoods in its interior. This only restricts the ambient chart. Write the two oriented handles as \begin{equation} H_r=[-2,2]\times D^2, \qquad \{-2,2\}\times D^2\subset\Int(Q), \qquad r=1,2, \label{eq:appf-handle-model} \end{equation} with orientation \(dt\wedge dx\wedge dy\). Their co-cores and chosen positive normals are \begin{equation} D_r=\{0\}\times D^2, \qquad n_r=\partial_t. \label{eq:appf-cocores} \end{equation} The product coordinates on the lateral annuli \([-2,2]\times\partial D^2\) induce the framings used below. Let \(\varepsilon>0\) be the constant height of the prescribed positive product push-off. After rescaling the handle coordinate, without moving the push-off, assume \(\varepsilon<1/10\) and write \begin{equation} D_2^{\mathrm{po}}=\{\varepsilon\}\times D^2\subset H_2. \label{eq:appf-positive-pushoff} \end{equation} Thus \(D_2^{\mathrm{po}}\) is the push-off in the direction \(n_2\), and it is disjoint from \(D_1\cup D_2\). Let \(a_\gamma\) be the oriented core of the prescribed band, including its short lateral-annulus segments. The hypotheses of \cref{thm:positive-two-site-input} put its first endpoint on \(m_1=\partial D_1\), its second on \(\partial D_2^{\mathrm{po}}\), away from the two diagonal dual-row points, and its interior away from both boundary circles and every other protected marking. Parametrize the already prescribed high-boundary rectangle by \(\bar\gamma=[-1,1]\times[0,1]\) with its product orientation and its short sides at \(v=0,1\). The supplied boundary-parallel push-in datum is a smooth graph in the frozen inward collar: for its recorded function \(b_\gamma\), that same prescribed band is \begin{equation} (u,v)\longmapsto \bigl(\bar\gamma(u,v),s=b_\gamma(u,v)\bigr), \qquad 0\le b_\gamma<\varepsilon/4,\qquad b_\gamma(u,0)=b_\gamma(u,1)=0. \label{eq:appf-band-pushin} \end{equation} The graph is unchanged at the attaching short sides, and its push-in datum carries the constant product normal of the boundary rectangle to the supplied framing. Thus \eqref{eq:appf-band-pushin} records the fixed band \(\gamma\), rather than constructing a new one. Its supplied orientation makes the boundary band sum positive. \begin{lemma}[Small slide ball] \label{lem:appf-small-slide-ball} The regular neighborhood \[ B_\gamma^{\mathrm{sl}} =\overline{N(D_1\cup D_2^{\mathrm{po}}\cup a_\gamma)} \] is a closed \(3\)-ball. Its three inward disk faces are product-parallel copies of \(D_1\), \(D_2^{\mathrm{po}}\), and the smoothed oriented disk \[ D_1^\gamma=D_1\natural_\gamma D_2^{\mathrm{po}}. \] The last disk is embedded, its boundary is the positive band sum, and it is disjoint from \(D_2\). We put \(D_2^\gamma=D_2\). \end{lemma} \begin{proof} The two disks are disjoint, and the interior of \(a_\gamma\) misses both. The regular-neighborhood disk-slide construction \cite[Section~4, disk-slide construction]{GhigginiLisca2014} identifies their neighborhood with a \(3\)-ball; the complement of its high-boundary pair of pants consists of the three stated disk faces. The third is obtained by deleting the two attaching intervals and adjoining the long sides of the rectangle \eqref{eq:appf-band-pushin}. In each prescribed band-corner smoothing box take adapted oriented quadrant coordinates \((r,s)\in[0,1]^2\) in which its fixed rounding is represented by \(r^2+s^2=\eta^2\), \(r,s\ge0\), for one common recorded \(\eta\). The four supplied smoothing boxes are disjoint. The result is a disk, because it is a boundary connected sum of two disks, and is embedded, because the disk pieces, band interior, and smoothing boxes are mutually disjoint. Its boundary orientation is the chosen positive band-sum orientation, so \begin{equation} [\partial D_1^\gamma]=[m_1]+[m_2]. \label{eq:appf-positive-meridian} \end{equation} Finally, take the regular neighborhood smaller than the positive distance from \(D_1\cup D_2^{\mathrm{po}}\cup a_\gamma\) to \(D_2\). This proves the asserted disjointness. Notice that the frontier of this small ball contains both an old and a slid disk face. Thus this ball constructs \(D_1^\gamma\), but it cannot be the support of a map which is fixed on its whole frontier. Only the stated local regular-neighborhood conclusion is used; the subsequent absolute minimal-disk-system assertion in the cited source is not imported. \end{proof} \begin{lemma}[Completeness of the two disk systems] \label{lem:appf-relative-completeness} The ordered systems \((D_1,D_2)\) and \((D_1^\gamma,D_2^\gamma)\) are complete relative co-core systems for the same two-handle compression body. Cutting the entire compression body along either pair produces the underlying product compression body, not a \(3\)-ball. \end{lemma} \begin{proof} Cutting \(H_r\) at \(t=0\) gives the half-handles \([-2,0]\times D^2\) and \([0,2]\times D^2\). Each is absorbed into the product collar on its attaching foot. Cutting both handles therefore recovers the product region, relative to the incoming boundary and the complement of the four foot collars. For the slid pair, first cut along \(D_2\). The strip between \(D_2=\{0\}\times D^2\) and \(D_2^{\mathrm{po}}=\{\varepsilon\}\times D^2\) then becomes a boundary product \(D^2\times[0,\varepsilon]\). Push the \(D_2^{\mathrm{po}}\)-lobe of \(D_1^\gamma\), together with the adjacent half of the band, monotonically through this product. At time \(u\), move the slice \(D^2\times\{\varepsilon(1-u)\}\) toward the attaching short side. At \(u=1\) the lobe and the two adjacent band strips cancel, leaving \(D_1\). The four corner roundings move inside their disjoint quadrant boxes, so this is a proper isotopy with no self-intersection. It is fixed off \(B_\gamma^{\mathrm{sl}}\) and on the incoming boundary. Cutting the remaining disk therefore gives the same product region as the old pair. \end{proof} \subsection{The larger sector and the two cut balls} Let \(C_\gamma\) be the closed product-ended piece cut off by the supplied interface \(E_\gamma\). In an adapted product coordinate, rescaled without moving the interface, write \(C_\gamma=Q\times[1-\tau,1]\) with \(0<\tau<1/10\). Its rounded frontier against the unchanged product region is precisely the prescribed disk \begin{equation} E_\gamma= \bigl(Q\times\{1-\tau\}\bigr) \cup\bigl(\partial Q\times[1-\tau,1]\bigr). \label{eq:appf-interface-disk} \end{equation} Let \(U_\gamma\) be obtained from \(C_\gamma\) by adjoining both complete handles. The interface hypothesis places the prescribed bands strictly inside this sector. Shrinking only the auxiliary regular neighborhood if necessary gives \(B_\gamma^{\mathrm{sl}}\subset U_\gamma\). \begin{lemma}[Sector and cut balls] \label{lem:appf-sector-cut-balls} The sector \(U_\gamma\) is a genus-two handlebody and meets the closure of its complement exactly in \(E_\gamma\). Its exposed boundary \[ P=\overline{\partial U_\gamma\setminus E_\gamma} \] is a genus-two surface with one boundary component \(\delta=\partial E_\gamma\). Both cut manifolds \begin{equation} K_{\mathfrak C} =U_\gamma\mathbin{\backslash\!\backslash}(D_1\cup D_2), \qquad K_{\mathfrak C^\gamma} =U_\gamma\mathbin{\backslash\!\backslash} (D_1^\gamma\cup D_2^\gamma) \label{eq:appf-two-cut-balls} \end{equation} are \(3\)-balls. Their boundaries contain, respectively, \[ D_1^-,D_1^+,D_2^-,D_2^+, \qquad (D_1^\gamma)^-,(D_1^\gamma)^+, (D_2^\gamma)^-,(D_2^\gamma)^+, \] and both contain the same interface disk \(E_\gamma\). \end{lemma} \begin{proof} The piece \(C_\gamma\) is a \(0\)-handle. Attaching the two products \([-2,2]\times D^2\) along their pairs of feet adds two \(1\)-handles, so \(U_\gamma\) is a genus-two handlebody. Its explicit product description shows that its only interface with the complement is \eqref{eq:appf-interface-disk}. Removing this disk from the boundary of a genus-two handlebody gives the stated surface \(P\), together with the directed collar of \(\delta\) inherited from the rounded coordinates. After cutting the old disks, \(C_\gamma\) has four product half-handles attached along boundary disks. Attaching a ball to a ball along a disk is again a ball: in collar coordinates the attached piece is compressed by \((z,s)\mapsto(z,s/2)\). Applying this four times proves that \(K_{\mathfrak C}\) is a ball and identifies the four signed faces. For the slid cut, first cut \(D_2\) and use the inverse-lobe isotopy from \cref{lem:appf-relative-completeness}; it takes \(D_1^\gamma\) to \(D_1\) in the cut sector and fixes \(E_\gamma\). The remaining cut is then the same four-half-handle construction. This proves independently that the second cut manifold is a ball. \end{proof} The balls in \eqref{eq:appf-two-cut-balls}, rather than the small ball \(B_\gamma^{\mathrm{sl}}\), are the domain and target of the extension. \subsection{The embedded dual row} In \(P\), retain the prescribed dual band \(\gamma^\vee\): the narrow boundary rectangle parallel to the old product band, running from the \(\ell_2\)-side to a positive push-off of the \(\ell_1\)-side on the specified side of \(\bar\gamma\), with its second short side oriented oppositely to \(\ell_1\). Its supplied interior is disjoint from all four old curves away from its ends. Define \begin{equation} \ell_1^\gamma=\ell_1, \qquad \ell_2^\gamma =\ell_2\natural_{\gamma^\vee}(-\ell_1)^+. \label{eq:appf-dual-band} \end{equation} \begin{lemma}[Geometric rows and signs] \label{lem:appf-geometric-rows} The four slid curves are embedded, oriented, and product framed. Their oriented classes are \begin{equation} \begin{aligned} [m_1^\gamma]&=[m_1]+[m_2],& [\ell_1^\gamma]&=[\ell_1],\\ [m_2^\gamma]&=[m_2],& [\ell_2^\gamma]&=[\ell_2]-[\ell_1]. \end{aligned} \label{eq:appf-four-classes} \end{equation} After the prescribed relative bigon removal, the actual intersections are \begin{equation} m_r^\gamma\mathbin{\cdot}\ell_s^\gamma=\delta_{rs}, \qquad m_r^\gamma\cap\ell_s^\gamma=\varnothing\quad(r\ne s), \qquad m_1^\gamma\cap m_2^\gamma=\varnothing, \qquad \ell_1^\gamma\cap\ell_2^\gamma=\varnothing. \label{eq:appf-geometric-pairing} \end{equation} Cutting along either disk system turns each longitude into an oriented arc from the negative to the positive copy of its disk face; its product framing becomes a ribbon neighborhood. \end{lemma} \begin{proof} The boundary components of the meridian pair of pants are \(m_1,m_2,-m_1^\gamma\); those of the longitude pair of pants are \(\ell_2,-\ell_1,-\ell_2^\gamma\). Their oriented boundary relations give \eqref{eq:appf-four-classes}, including its two independent signs. Retain the original positive points \(m_1\cap\ell_1\) and \(m_2\cap\ell_2\) outside the band neighborhoods. Initially \(m_1^\gamma\) meets \(\ell_2^\gamma\) twice: the \(m_2\)-lobe meets the \(\ell_2\)-piece positively, while the \(m_1\)-piece meets the \(( -\ell_1)^+\)-piece negatively. Parallel routing of the two bands makes these points the corners of an embedded bigon disjoint from every other curve. Pushing the longitude side across this bigon removes both points and preserves the product framing. All other cross pairs are disjoint by construction, proving \eqref{eq:appf-geometric-pairing} geometrically. Cutting at a longitude--co-core crossing separates a small interval into endpoints on the two signed faces; cutting its framed normal interval at the same time gives the stated ribbon. \end{proof} \subsection{A complete dictionary on the two cut spheres} \label{subsec:appf-cell-dictionary} We now make the boundary map finite and explicit. Write \(\mathsf e_\gamma\) for the cell vertex representing the local interface disk \(E_\gamma\); this is not the global interface symbol \(E\) of \cref{def:supplied-protected-sector}. On the old sphere use the five disk vertices \[ \mathcal V^{\mathrm{o}} =\{\mathsf e_\gamma,1-,1+,2-,2+\}; \] on the slid sphere use \[ \mathcal V^\gamma =\{\mathsf e_\gamma,1\gamma-,1\gamma+,2\gamma-,2\gamma+\}. \] Let \(\nu:\mathcal V^{\mathrm{o}}\to\mathcal V^\gamma\) be the displayed index-preserving bijection. The oriented fat-tree edges are \begin{equation} s_1:\mathsf e_\gamma\longrightarrow1-, \quad c_1:1-\longrightarrow1+, \quad s_2:\mathsf e_\gamma\longrightarrow2-, \quad c_2:2-\longrightarrow2+, \label{eq:appf-tree} \end{equation} and the target has the correspondingly decorated four edges. The edges \(c_1,c_2\) are the actual dual arcs from \cref{lem:appf-geometric-rows}; \(s_1,s_2\) are auxiliary arcs chosen in the complement of their ribbons. Put \[ n_{\mathsf e_\gamma}=n_{1-}=n_{2-}=4, \qquad n_{1+}=n_{2+}=2. \] For every \(v\in\mathcal V^{\mathrm{o}}\) and \(1\le j\le n_v\), with \(j\) read cyclically, take vertices \(x_{v,j}^{\mathrm{o}},y_{v,j}^{\mathrm{o}}\), inner edges \[ \iota_{v,j}^{\mathrm{o}}: x_{v,j}^{\mathrm{o}}\longrightarrow x_{v,j+1}^{\mathrm{o}}, \] outer edges \[ \omega_{v,j}^{\mathrm{o}}: y_{v,j}^{\mathrm{o}}\longrightarrow y_{v,j+1}^{\mathrm{o}}, \] and radial edges \[ \rho_{v,j}^{\mathrm{o}}: x_{v,j}^{\mathrm{o}}\longrightarrow y_{v,j}^{\mathrm{o}}. \] Use the same definitions with superscript \(\gamma\) and vertex \(\nu(v)\) on the target sphere. This is a complete list of the thirty-two vertices and the forty-eight inner, outer, and radial edges on either sphere. The attachment names for the outer edges are fixed by \begin{equation} \begin{array}{c|cccc} v&\omega_{v,1}&\omega_{v,2}&\omega_{v,3}&\omega_{v,4}\\ \hline \mathsf e_\gamma& a_{\mathsf e_\gamma,s_1}&f_{\mathsf e_\gamma,0}& a_{\mathsf e_\gamma,s_2}&f_{\mathsf e_\gamma,1}\\ 1-&a_{1-,s_1}&f_{1-,0}&a_{1-,c_1}&f_{1-,1}\\ 2-&a_{2-,s_2}&f_{2-,0}&a_{2-,c_2}&f_{2-,1} \end{array} \qquad \begin{array}{c|cc} v&\omega_{v,1}&\omega_{v,2}\\ \hline 1+&a_{1+,c_1}&f_{1+}\\ 2+&a_{2+,c_2}&f_{2+} \end{array}. \label{eq:appf-outer-edge-names} \end{equation} The target table is obtained by applying \(\nu\) to every non-\(\mathsf e_\gamma\) vertex and decorating each tree-edge name. For every tree edge \(e\), add two long edges \(l_e,r_e\), directed from its initial to its terminal vertex. These eight long edges bring the total to fifty-six. The following formulas list all twenty-six faces and all of their cyclic words. They apply once to the old data and once to the decorated target data: \begin{align} F_v&: \iota_{v,1}\iota_{v,2}\cdots\iota_{v,n_v}, &&v\in\mathcal V^{\mathrm{o}}, \label{eq:appf-core-face-words}\\ C_{v,j}&: \rho_{v,j}\omega_{v,j}\rho_{v,j+1}^{-1}\iota_{v,j}^{-1}, &&v\in\mathcal V^{\mathrm{o}},\ 1\le j\le n_v, \label{eq:appf-collar-face-words}\\ R_e&: l_ea_{\operatorname{ter}(e),e}^{-1} r_e^{-1}a_{\operatorname{ini}(e),e}^{-1}, &&e\in\{s_1,c_1,s_2,c_2\}. \label{eq:appf-ribbon-face-words} \end{align} Here the five \(F_v\) are the interface disk and four signed disk faces, the sixteen \(C_{v,j}\) are their full collar sectors, and the four \(R_e\) are the tree ribbons. The remaining face \(Z\) has the following complete sixteen-edge boundary word: \begin{equation} \begin{split} &(f_{\mathsf e_\gamma,0})^{-1}r_{s_1} (f_{1-,1})^{-1}r_{c_1} (f_{1+})^{-1}l_{c_1}^{-1}(f_{1-,0})^{-1}l_{s_1}^{-1}\\ &\qquad\cdot (f_{\mathsf e_\gamma,1})^{-1}r_{s_2} (f_{2-,1})^{-1}r_{c_2} (f_{2+})^{-1}l_{c_2}^{-1}(f_{2-,0})^{-1}l_{s_2}^{-1}. \end{split} \label{eq:appf-complement-word} \end{equation} For the target, decorate this word as above. Its successive source vertices are \begin{equation} \begin{split} y_{\mathsf e_\gamma,3}&,y_{\mathsf e_\gamma,2}, y_{1-,1},y_{1-,4},y_{1+,1},y_{1+,2},y_{1-,3},y_{1-,2}, y_{\mathsf e_\gamma,1},\\ &y_{\mathsf e_\gamma,4},y_{2-,1},y_{2-,4}, y_{2+,1},y_{2+,2},y_{2-,3},y_{2-,2}, y_{\mathsf e_\gamma,3}. \end{split} \label{eq:appf-complement-vertices} \end{equation} Thus every consecutive pair, including the last and first edges, shares the indicated endpoint. For clarity, the source--target product dictionary is: \begin{center} \begin{tabularx}{\textwidth}{@{}>{\raggedright\arraybackslash}p{0.17\textwidth} >{\raggedright\arraybackslash}p{0.22\textwidth} >{\raggedright\arraybackslash}p{0.29\textwidth}>{\raggedright\arraybackslash}X@{}} \toprule faces & target & prescribed map & pairing or role\\ \midrule \(F_{\mathsf e_\gamma}^{\mathrm{o}}\) & \(F_{\mathsf e_\gamma}^\gamma\) & identity in the fixed disk chart \(d_{\mathsf e_\gamma}\) and its normal collar & interface\\ \(F_{r-}^{\mathrm{o}}\) & \(F_{r\gamma-}^\gamma\) & \(d_{r-}^{\mathrm{o}}\times t\) to \(d_{r\gamma-}^\gamma\times t\) & paired with \(F_{r+}\)\\ \(F_{r+}^{\mathrm{o}}\) & \(F_{r\gamma+}^\gamma\) & \(d_{r+}^{\mathrm{o}}\times t\) to \(d_{r\gamma+}^\gamma\times t\) & paired with \(F_{r-}\)\\ \(C_{v,j}^{\mathrm{o}}\) & \(C_{\nu(v),j}^\gamma\) & \(\kappa_{\nu(v),j}^\gamma\circ{\kappa_{v,j}^{\mathrm{o}}}^{-1}\) & pointwise product map on the entire collar\\ \(R_e^{\mathrm{o}}\) & \(R_e^\gamma\) & \(\varrho_e^\gamma\circ{\varrho_e^{\mathrm{o}}}^{-1}\) & actual dual ribbon for \(e=c_r\), auxiliary ribbon for \(e=s_r\)\\ \(Z^{\mathrm{o}}\) & \(Z^\gamma\) & indexed affine map after coning & sole unmarked complementary disk\\ \bottomrule \end{tabularx} \end{center} In the two signed-face rows \(r=1,2\); in the collar row \(v\in\mathcal V^{\mathrm{o}}\) and \(1\le j\le n_v\); and in the ribbon row \(e\in\{s_1,c_1,s_2,c_2\}\). These displayed finite ranges are part of the dictionary. Every inner edge occurs once in a core word and inversely in a collar word. Every radial edge occurs oppositely in two adjacent collar words; every attachment edge occurs in a collar and inversely in a ribbon; every free outer edge occurs in a collar and inversely in \(Z\); and every long edge occurs in a ribbon and inversely in \(Z\). Thus each of the fifty-six edges occurs twice with opposite orientations. There are thirty-two vertices and twenty-six faces, and \begin{equation} 32-56+26=2. \label{eq:appf-euler-check} \end{equation} The word \eqref{eq:appf-complement-word} is the boundary walk of a regular neighborhood of the tree \eqref{eq:appf-tree}. At each vertex, the core, collar, ribbon, and complementary corners occur once in the displayed cyclic order. Hence every vertex link is a circle. The connected complex is therefore a closed oriented surface of Euler characteristic two, so it is the whole cut sphere; in particular \(Z\) is a disk and no face is missing. Put an interior vertex \(z^{\mathrm{o}}\) in \(Z^{\mathrm{o}}\) and \(z^\gamma\) in \(Z^\gamma\), and cone the sixteen edges in \eqref{eq:appf-complement-word}. Denote the resulting triangles by \(T_j^{\mathrm{o}}\) and \(T_j^\gamma\), \(1\le j\le16\), in boundary order. There are no other complementary triangles. The four \(\mathsf e_\gamma\)-collar rectangles carry coordinates \begin{equation} (\theta,u)\in(\mathbb R/\mathbb Z)\times[-1,1], \qquad \theta=0,\tfrac14,\tfrac12,\tfrac34 \label{eq:appf-delta-coordinates} \end{equation} on their radial edges. Choose the trace of the detector \(b_\delta\) to be \(\rho_{\mathsf e_\gamma,1}\), directed from \(u=-1\) on the \(E_\gamma\)-side to \(u=1\) on the \(P\)-side. The union \(C_{\mathsf e_\gamma,4}\cup C_{\mathsf e_\gamma,1}\) is its framed ribbon. Its two outside radial edges record the normal and product framing, and its incident inner and outer edges record both endpoint and tangent germs. The rectangles \(R_{c_1},R_{c_2}\) are precisely the two geometric dual-arc ribbons. Thus the cellulation contains every signed disk face, dual arc, framing ribbon, product collar, detector trace, and auxiliary cutting arc. \begin{proposition}[Complete cut-sphere map] \label{prop:appf-cut-sphere-map} There is an orientation-preserving cellular homeomorphism \[ \phi_\gamma^{\mathrm{PL}}: (\partial K_{\mathfrak C},\mathcal Q_{\mathfrak C}) \longrightarrow (\partial K_{\mathfrak C^\gamma}, \mathcal Q_{\mathfrak C^\gamma}) \] which is the identity on \(E_\gamma\) and its product germ, maps every signed old disk face and its complete normal collar to the corresponding slid face, maps the actual old dual arcs and ribbons to the actual slid ones, and is the prescribed pointwise product map on every collar. It intertwines each source minus-to-plus pairing with its corresponding target pairing, and therefore also intertwines their inverses. On the \(\delta\)-collar it is \((\theta,u)\mapsto(\theta,u)\), so the detector has zero turns. \end{proposition} \begin{proof} The complete vertex and edge assignment is, for every displayed finite index, \begin{align*} x_{v,j}^{\mathrm{o}}&\longmapsto x_{\nu(v),j}^\gamma& y_{v,j}^{\mathrm{o}}&\longmapsto y_{\nu(v),j}^\gamma,\\ \iota_{v,j}^{\mathrm{o}}&\longmapsto\iota_{\nu(v),j}^\gamma& \omega_{v,j}^{\mathrm{o}}&\longmapsto\omega_{\nu(v),j}^\gamma& \rho_{v,j}^{\mathrm{o}}&\longmapsto\rho_{\nu(v),j}^\gamma,\\ l_e^{\mathrm{o}}&\longmapsto l_e^\gamma& r_e^{\mathrm{o}}&\longmapsto r_e^\gamma. \end{align*} Map the five disk cores, sixteen collar rectangles, and four ribbons by the product maps in the dictionary. Send \(z^{\mathrm{o}}\) to \(z^\gamma\) and map \(T_j^{\mathrm{o}}\) affinely and orientation preservingly to \(T_j^\gamma\), for each \(1\le j\le16\). The cyclic words show that adjacent formulas agree on every common edge. The map is bijective on vertices, edges, and faces and preserves every face orientation, so it is an orientation-preserving homeomorphism of the whole spheres. Let \(j_r^{\mathrm{o}}\) and \(j_r^\gamma\) be the orientation-reversing pairings of the two signed copies of the \(r\)-th disk. The paired disk maps use the same normal-side coordinate, whence \begin{equation} \phi_\gamma^{\mathrm{PL}}j_r^{\mathrm{o}}(z) =j_r^\gamma\phi_\gamma^{\mathrm{PL}}(z), \qquad r=1,2. \label{eq:appf-pl-pairing} \end{equation} The \(\mathsf e_\gamma\)-rows are literal identities, and the \(c_r\)-rows are the actual geometric ribbons of \cref{lem:appf-geometric-rows}. Finally, \eqref{eq:appf-delta-coordinates} is preserved pointwise, with no term \(\theta\mapsto\theta+qu\). Thus both endpoint germs, the directed tangent, the chosen normal, and the two framing edges of \(b_\delta\) are transported with \(q=0\). \end{proof} \subsection{A smooth boundary representative} The cellular map fixes the incidence and relative class, but agreement of affine values along a cell edge is not yet agreement of smooth germs. We next construct the smooth boundary map which will be extended over the balls. The smooth representative is built in the same order as the cellular one. First choose compatible vertex and edge charts, then use the prescribed product maps on the protected faces and ribbons. The only free piece is the complementary disk: an interpolation of the boundary circle map fills it while remaining the identity near its center. The resulting sphere map has the same full annular germ as the cellular map, and the Alexander family records that the two fillings lie in the same topological relative class. Let \(\mathcal P_\partial^{\mathrm{o}}\) be a closed regular neighborhood of the first twenty-five faces in the dictionary: it contains the interface and its collar, all signed disk faces and collars, all four ribbons, the inherited site collars, the directed \(\delta\)-collar, and the full framed detector trace. Its complementary face is the single disk \(Z_0^{\mathrm{o}}\). Round the finitely many corners of the common boundary inside the prescribed face collars and adjoin to \(Z_0^{\mathrm{o}}\) a smaller closed product collar of that rounded boundary. Denote the enlarged disk by \(Z^{\mathrm{o}}\) and the overlap annulus by \[ A^{\mathrm{o}} =\mathcal P_\partial^{\mathrm{o}}\cap Z^{\mathrm{o}}. \] Make the identical indexed construction in the target. Thus the two pieces cover each sphere and overlap in one full closed annulus, not merely in a circle. At every vertex in a prescribed product chart, use that chart and its indexed target chart. At every remaining vertex choose a small oriented smooth disk chart, and choose the target chart simultaneously so that the local indexed map is the coordinate identity. Make these vertex disks disjoint. For each open edge, remove the charted endpoint intervals and choose a two-sided tubular chart \[ (s,n)\in[a,b]\times[-\varepsilon_e,\varepsilon_e] \] whose \(s\)-coordinate agrees with the endpoint charts. On a frozen edge this is the prescribed product chart; on any other edge choose the target chart by transporting the source chart through its indexed parameterization. Shrinking first the finitely many vertex disks and then the edge strips makes every overlap the same coordinate expression \((s,n)\mapsto(s,n)\). The one-skeleton map is therefore smooth with one two-sided germ at every vertex and edge. On the five core faces use the frozen disk parameterizations, on the sixteen collar faces their pointwise product parameterizations, and on the four ribbon faces their oriented rectangle parameterizations. Their boundary variables are exactly the variables just fixed on the one-skeleton. Adjacent face germs hence agree on a full two-sided strip. For a paired pair of signed disk faces, use the same normal variable and define the positive-face map from the negative-face map by conjugating with \(j_r^{\mathrm{o}}\) and \(j_r^\gamma\). This gives a smooth orientation-preserving diffeomorphism \[ \phi_{\gamma,\mathcal P}^{\mathrm{sm}}: \mathcal P_\partial^{\mathrm{o}} \longrightarrow\mathcal P_\partial^\gamma \] which preserves every protected disk, ribbon, collar, tangent, normal, and framing in its actual product coordinates. It remains to fill the sole disk \(Z^{\mathrm{o}}\). Choose the source and target smooth disk parameterizations simultaneously with their collars. For some \(00. \] For \(0\le s\le1\), put \[ f_s(\theta)=(1-s)\theta+s f(\theta). \] Then \(\partial_\theta f_s=(1-s)+sf'(\theta)>0\), so each \(f_s\) is a circle diffeomorphism. Choose \(00\), its angular Jacobian is \((1-\eta(r))+\eta(r)f'(\theta)>0\); it preserves \(r\), and hence is an orientation-preserving disk diffeomorphism. Its inverse is obtained radius by radius from the smooth inverses of the circle maps \(f_{\eta(r)}\); it is the identity near the center and the fixed product inverse near the annulus. Endpoint flatness gives agreement to all orders at the two radial seams. The protected and complementary formulas glue to an orientation-preserving sphere diffeomorphism \begin{equation} \phi_\gamma^{\mathrm{sm}}: \partial K_{\mathfrak C}\longrightarrow \partial K_{\mathfrak C^\gamma}. \label{eq:appf-smooth-sphere-map} \end{equation} It represents the cellular relative class. Indeed, on the inner disk bounded by \(A^{\mathrm{o}}\), let \(p\) be the cellular filling and \(q\) the smooth filling; they have the same boundary map. Put \(h=q^{-1}p\) and define \begin{equation} H_s(x)= \begin{cases} s h(x/s),&|x|\le s,\\ x,&s\le|x|\le1, \end{cases} \quad 00\) and full inward smooth collars \[ c^{\mathrm{o}}:\partial K_{\mathfrak C} \times[0,3\epsilon_{\mathrm{col}}] \longrightarrow K_{\mathfrak C}, \qquad c^\gamma:\partial K_{\mathfrak C^\gamma} \times[0,3\epsilon_{\mathrm{col}}] \longrightarrow K_{\mathfrak C^\gamma}, \] extending every fixed face-normal coordinate. On \(0\le t\le2\epsilon_{\mathrm{col}}\), prescribe \begin{equation} c^{\mathrm{o}}(x,t)\longmapsto c^\gamma(\phi_\gamma^{\mathrm{sm}}(x),t). \label{eq:appf-full-collar-product} \end{equation} The complements of the open collars \(t<2\epsilon_{\mathrm{col}}\) are closed balls. Parameterize them by standard balls so that the boundary coordinates are the \(t=2\epsilon_{\mathrm{col}}\) copies above. If \(\psi_\partial:S^2\to S^2\) is the resulting inner boundary map, extend it radially by \begin{equation} \mathcal A_{\psi_\partial}(ru)=r\psi_\partial(u), \qquad 0\le r\le1,\quad u\in S^2, \label{eq:appf-radial-ball-extension} \end{equation} with \(\mathcal A_{\psi_\partial}(0)=0\). This is continuous at the center, and its inverse is \(\mathcal A_{\psi_\partial^{-1}}\). It agrees with \eqref{eq:appf-full-collar-product} at the inner seam. The two formulas therefore give a homeomorphism \begin{equation} \Phi_\gamma^{\mathrm A}: K_{\mathfrak C}\longrightarrow K_{\mathfrak C^\gamma} \label{eq:appf-alexander-homeomorphism} \end{equation} which is a smooth orientation-preserving diffeomorphism on the full outer collar \(0\le t<\epsilon_{\mathrm{col}}\). We do not claim that the radial formula is smooth at its cone point. Double the two cut balls along their boundaries. The signed collar coordinate identifies a neighborhood of each equator with \(S^2\times(-\epsilon_{\mathrm{col}},\epsilon_{\mathrm{col}})\), and the doubled map \(\widehat\Phi_\gamma^{\mathrm A}\) has the smooth product formula \[ (x,t)\longmapsto(\phi_\gamma^{\mathrm{sm}}(x),t) \] there. Let \(K\) be the equator together with the closed thinner two-sided collar \[ S^2\times [-\epsilon_{\mathrm{col}}/2,\epsilon_{\mathrm{col}}/2]. \] This closed set lies inside the open region on which \(\widehat\Phi_\gamma^{\mathrm A}\) is already a diffeomorphism. The classical relative homeomorphism-approximation theorem, in the form stated by Pardon \cite[Theorem~4.6]{Pardon2020}, applied directly to \(\widehat\Phi_\gamma^{\mathrm A}\), supplies a diffeomorphism \(\widehat\Psi_\gamma\) of the source double to the target double which equals \(\widehat\Phi_\gamma^{\mathrm A}\) on a neighborhood of \(K\). The equator separates each double into two connected components. A point of the selected source side near the equator maps by the fixed product formula to the correspondingly labelled target side. Connectedness puts the image of the whole source side in that target side; applying the same argument to \(\widehat\Psi_\gamma^{-1}\) proves surjectivity onto it. Restricting to the closures of the chosen sides therefore gives a diffeomorphism \begin{equation} \widetilde\Phi_\gamma: K_{\mathfrak C}\longrightarrow K_{\mathfrak C^\gamma} \label{eq:appf-smooth-cut-map} \end{equation} which agrees with \eqref{eq:appf-full-collar-product} on a full boundary collar. The smooth cut map lies in the intended topological relative class. In dimension \(n=3\), apply the same formula \begin{equation} H_s(x)= \begin{cases} s h(x/s),&|x|\le s,\\ x,&s\le|x|\le1, \end{cases} \quad 0