\section{Five cut schedules and complete cellulations} \label{app:cut-schedules} This appendix records the finite disk-and-cell data used in \cref{sec:vertex-geometry}. The purpose of printing the complete lists is twofold: the terminal nested cancellation cannot be replaced by a simultaneous isotopy, and the cut-sphere realization requires the actual oriented face cycles rather than an Euler-characteristic count. \subsection{The product-germ cancellation mechanism} Consider a product \(1\)-handle \([-2,2]\times D^2\) with middle co-core \(D_s=\{0\}\times D^2\). Suppose that a second disk has a lobe on a positive parallel copy \(D_s^+\), joined to its root disk by a product-zero band in a supplied rectangular box. Cutting along \(D_s\) turns the handle into two half-handles. In the half-handle containing \(D_s^+\), the lobe and its band are boundary parallel: in product coordinates their union is the graph of an embedded strip whose short side lies on the new cut face. Compress this strip monotonically toward that short side. Extending the compression by the identity in the product-normal variable produces an ambient isotopy of the already cut manifold. It removes the lobe and retracts the disk to its root while preserving its full normal germ, the global \(E\)-bicollar, and every disjoint marking germ. If two lobes occupy ordered heights in the same handle, this mechanism is applied from the nearer lobe to the farther one. The first isotopy clears the track used by the second. Nothing in the construction asserts that the two sweeps have disjoint simultaneous supports. \subsection{The five disk rows and cut schedules} For reference, the row disks are \begin{equation} \begin{array}{c|ccc} v&D_i^v&D_j^v&D_k^v\\ \hline 0&i_0&j_0&k_0\\ B&i_0&J&k_0\\ A&I_A&j_0&k_0\\ AC&I_\star&j_0&k_0\\ \starv&I_\star&J&k_0, \end{array} \qquad \begin{aligned} J&=j_0\#_{\beta_0}k_\varepsilon,\\ I_A&=i_0\#_\alpha j_\varepsilon,\\ I_\star&=i_0\#_\alpha J^+ =(I_A)\#_{\beta_1}k_{2\varepsilon}. \end{aligned} \label{eq:appendix-five-disk-rows} \end{equation} In a schedule below, \(\operatorname{cut}D\) means cut the actual disk in the displayed row; a following arrow describes its image after the earlier cuts. \begin{table}[htbp] \centering \small \caption{The five product-germ-preserving cut schedules.} \label{tab:five-cut-schedules} \begin{tabularx}{\textwidth}{c>{\raggedright\arraybackslash}X} \hline vertex&operations in the already cut manifold\\ \hline \(0\)& \(\operatorname{cut}k_0\); then \(\operatorname{cut}j_0\); then \(\operatorname{cut}i_0\).\\[0.3em] \(B\)& \(\operatorname{cut}k_0\); \(J\rightsquigarrow j_0\) by removing its \(k_\varepsilon\)-lobe; \(\operatorname{cut}J\) (now represented by \(j_0\)); \(\operatorname{cut}i_0\).\\[0.3em] \(A\)& \(\operatorname{cut}j_0\); \(I_A\rightsquigarrow i_0\) by removing its \(j_\varepsilon\)-lobe; \(\operatorname{cut}I_A\) (now represented by \(i_0\)); \(\operatorname{cut}k_0\).\\[0.3em] \(AC\)& \(\operatorname{cut}k_0\); \(I_\star\rightsquigarrow I_A\) by removing the \(k_{2\varepsilon}\)-lobe and adjacent \(\beta_1\)-strips; \(\operatorname{cut}j_0\); \(I_A\rightsquigarrow i_0\); \(\operatorname{cut}I_\star\) (now represented by \(i_0\)).\\[0.3em] \(\starv\)& \(\operatorname{cut}k_0\); first remove the nearer \(k_\varepsilon\)-lobe so \(J\rightsquigarrow j_0\); then move the farther \(k_{2\varepsilon}\)-lobe of \(I_\star\) through the cleared region so \(I_\star\rightsquigarrow I_A\); \(\operatorname{cut}J\) (now represented by \(j_0\)); remove the \(j_\varepsilon\)-lobe so \(I_A\rightsquigarrow i_0\); \(\operatorname{cut}I_\star\) (now represented by \(i_0\)).\\ \hline \end{tabularx} \end{table} Each arrow is the product-germ cancellation described above. Its support lies in the supplied handle, band, and smoothing boxes and is disjoint from \(E\), the exterior row, and the unused handles. At \(AC\) the \(\beta_1\)-lobe is removed before the \(\alpha\)-lobe. At \(\starv\) the \(k_\varepsilon\)-lobe is removed before the \(k_{2\varepsilon}\)-lobe; the order is part of the schedule. After these cancellations every row reduces to the three elementary co-cores. Cutting them leaves the rounded central ball \[ C=\operatorname{Round}\bigl(\chi(Q\times[1-\tau,1])\bigr) \] together with six half-handles attached along boundary disks. Absorbing the half-handles by boundary-collar compression identifies the remainder with \(D^3\). Because every support misses the global interface, the identifications \begin{equation} \Phi_v:K_v\overset{\cong}{\longrightarrow}D^3, \qquad v\in V, \label{eq:appendix-relative-ball-identifications} \end{equation} are orientation preserving and fixed on the same \(E\)-bicollar. \subsection{The abstract oriented cut-sphere cellulation} The model has seven core disk faces \begin{equation} E,\quad i^-,i^+,\quad j^-,j^+,\quad k^-,k^+, \label{eq:appendix-seven-faces} \end{equation} and three branches \begin{equation} E\xrightarrow{s_r}r^-\xrightarrow{c_r}r^+, \qquad r=i,j,k. \label{eq:appendix-three-branches} \end{equation} Use six sectors on \(E\), four on every \(r^-\), and two on every \(r^+\). For a face \(f\) with \(n_f\) sectors, introduce vertices \(x_{f,a},y_{f,a}\) and oriented edges \begin{equation} \begin{aligned} a_{f,a}&:x_{f,a}\longrightarrow x_{f,a+1},\\ b_{f,a}&:y_{f,a}\longrightarrow y_{f,a+1},\\ d_{f,a}&:x_{f,a}\longrightarrow y_{f,a}, \end{aligned} \qquad a\in\mathbb Z/n_f. \label{eq:appendix-face-edge-convention} \end{equation} The spoke attachment slots on \(E\) are \begin{equation} p_i=1,\qquad p_j=5,\qquad p_k=3. \label{eq:appendix-spoke-slots} \end{equation} For each \(r\), orient the four long sides of the spoke and dual ribbons by the endpoint table \begin{equation} \begin{array}{c|c|c} \text{side}&\text{initial endpoint}&\text{terminal endpoint}\\ \hline l_{s_r}&y_{E,p_r}&y_{r^-,2}\\ r_{s_r}&y_{E,p_r+1}&y_{r^-,1}\\ l_{c_r}&y_{r^-,3}&y_{r^+,2}\\ r_{c_r}&y_{r^-,4}&y_{r^+,1}, \end{array} \label{tab:appendix-ribbon-sides} \end{equation} where the \(E\)-indices are cyclic. The seven oriented core faces are \begin{equation} \partial C_f=a_{f,1}a_{f,2}\cdots a_{f,n_f}. \label{eq:appendix-core-face-cycles} \end{equation} The \(n_f\) collar quadrilaterals at \(f\) are \begin{equation} \partial A_{f,a} =d_{f,a}\,b_{f,a}\,d_{f,a+1}^{-1}\,a_{f,a}^{-1}. \label{eq:appendix-collar-face-cycles} \end{equation} For every \(r=i,j,k\), the two ribbon quadrilaterals have boundary \begin{equation} \begin{aligned} \partial R_{s_r} &=l_{s_r}\,b_{r^-,1}^{-1}\,r_{s_r}^{-1}\, b_{E,p_r}^{-1},\\ \partial R_{c_r} &=l_{c_r}\,b_{r^+,1}^{-1}\,r_{c_r}^{-1}\, b_{r^-,3}^{-1}. \end{aligned} \label{eq:appendix-ribbon-face-cycles} \end{equation} Thus the six ribbons consume, on their non-collar sides, \begin{equation} b_{E,p_r},\quad b_{r^-,1},\quad b_{r^-,3},\quad b_{r^+,1} \qquad(r=i,j,k). \label{eq:appendix-consumed-outer-edges} \end{equation} On the two-sector face \(r^+\), the edges \(b_{r^+,1}\) and \(b_{r^+,2}\) have the same unordered endpoints but are distinct parallel \(1\)-cells. The ribbon uses the first and the complementary face uses the second. The remaining outer edges form the following simple oriented \(24\)-vertex cycle: \begin{align} y_{E,3}\to y_{E,2}\to y_{i^-,1}\to y_{i^-,4} \to y_{i^+,1}\to y_{i^+,2}\to y_{i^-,3}\to y_{i^-,2} \notag\\ {}\to y_{E,1}\to y_{E,6}\to y_{j^-,1}\to y_{j^-,4} \to y_{j^+,1}\to y_{j^+,2}\to y_{j^-,3}\to y_{j^-,2} \notag\\ {}\to y_{E,5}\to y_{E,4}\to y_{k^-,1}\to y_{k^-,4} \to y_{k^+,1}\to y_{k^+,2}\to y_{k^-,3}\to y_{k^-,2} \to y_{E,3}. \label{eq:appendix-complementary-24-cycle} \end{align} Every arrow is the unique incident oriented \(1\)-cell or its inverse, except at a positive face, where the indicated arrow is \(b_{r^+,2}\). Attaching one \(24\)-gon along \eqref{eq:appendix-complementary-24-cycle} completes the face list. \begin{proposition}[The finite model is a marked sphere] \label{prop:appendix-finite-sphere} The cycles \eqref{eq:appendix-core-face-cycles}--\eqref{eq:appendix-complementary-24-cycle} define a connected closed oriented cellulation of \(S^2\) with \begin{equation} (V,E,F)=(48,84,38). \label{eq:appendix-488438} \end{equation} The \(24\)-gon is the unique complementary disk. \end{proposition} \begin{proof} The incidence and link checks establish a connected closed oriented surface; only then does the Euler count identify it as a sphere. The fat-tree neighborhood identifies the \(24\)-gon as its complementary disk. There are \[ \begin{aligned} V&=2(6+3\cdot4+3\cdot2)=48,\\ E&=3(6+3\cdot4+3\cdot2)+3\cdot4=84,\\ F&=7+(6+3\cdot4+3\cdot2)+6+1=38. \end{aligned} \] The printed incidence list places every \(1\)-cell in exactly two face cycles with opposite induced orientations. Its one-skeleton is connected. Each \(x\)-vertex has a \(3\)-cycle link and each \(y\)-vertex a \(4\)-cycle link, including the explicitly distinguished parallel edge at \(r^+\). Hence the result is a connected closed orientable \(2\)-manifold. Its Euler characteristic is \(48-84+38=2\), so it is a sphere. Since all other faces lie in the regular neighborhood of the protected fat tree, the last face is its unique complementary disk. \end{proof} \subsection{Realized fat trees and all five vertex models} At vertex \(v\), denote the seven realized faces by \[ E,\quad D_{i,v}^-,D_{i,v}^+,\quad D_{j,v}^-,D_{j,v}^+,\quad D_{k,v}^-,D_{k,v}^+. \] The cut dual longitudes provide the three arcs \(c_{r,v}:D_{r,v}^-\to D_{r,v}^+\). Their union with the seven face disks has four components. Choose disjoint spokes \begin{equation} s_{r,v}:E\longrightarrow D_{r,v}^-, \qquad r=i,j,k, \label{eq:appendix-physical-spokes} \end{equation} one at a time in the supplied corridors, attaching to \(E\) in the slots \eqref{eq:appendix-spoke-slots}. Each joins two different components, so \begin{equation} T_v=(\text{seven face disks})\cup \bigcup_{r=i,j,k}(s_{r,v}\cup c_{r,v}) \label{eq:appendix-physical-fat-tree} \end{equation} is a connected fat tree. Its closed regular neighborhood is a disk and the closure of its complement in \(\partial K_v\) is a disk \(Z_v\). This proves the complementary-disk statement geometrically, before any Euler-characteristic check. Map the abstract faces, branches, ribbons, detector, and collar rectangles to these realized objects in their product coordinates. The coordinate maps agree on full overlap strips and give a positive diffeomorphism on a regular neighborhood of the tree. In polar charts on the two complementary disks, its boundary germ is \[ (r,[\theta])\longmapsto(r,[f_v(\theta)]), \quad f_v(\theta+1)=f_v(\theta)+1,\quad f_v'(\theta)>0. \] For an endpoint-flat cutoff \(\eta(r)\), extend it by \begin{equation} (r,[\theta])\longmapsto \bigl(r,[(1-\eta(r))\theta+\eta(r)f_v(\theta)]\bigr). \label{eq:appendix-polar-extension} \end{equation} Near \(r=0\) this is the identity, so there is no polar-center singularity. Away from the center its angular derivative is positive. It agrees with the tree-neighborhood map on an open outer annulus with all derivatives. The resulting \[ b_v:S^2_{Q0}\overset{\cong}{\longrightarrow}\partial K_v \] realizes every one of the \(38\) indexed faces. This construction is performed independently at all five vertices. \subsection{Signed faces and full normal pairings} For each \(r\), fix an orientation-reversing disk diffeomorphism \(\jmath_r:D^2\to D^2\). Choose a positive chart \(\kappa_{r,v}^{-,\partial}:D^2\to D_{r,v}^-\), and define \begin{equation} \kappa_{r,v}^{+,\partial} =\gamma_{r,v}\kappa_{r,v}^{-,\partial}\jmath_r^{-1}, \label{eq:appendix-positive-face-chart} \end{equation} where \(\gamma_{r,v}\) is the canonical identification of the two copies of the same row disk. Then \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:appendix-surface-pairing} \end{equation} The canonical identification and \(\jmath_r\) each reverse the relevant disk orientation, so both face charts are positive while the pairing is orientation reversing. It is a smooth pairing, not a cellular one: four displayed sectors on \(r^-\) cannot be paired cellwise with two on \(r^+\). Let \[ \nu_{r,v}^{\sigma}:D_{r,v}^{\sigma}\times[0,d_{r,v}) \longrightarrow K_v \] be the inherited inward collar. In its local double, use the formula \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|