\section{Slide data and local positive-slide insertion} \label{slide:sec-setup} \label{slide:sec:conventions-local-input} After all \(k\) labelled handles have been attached, a slide changes the marking of the completed compression body. Protected sectors specify where each local move is defined; their common collars allow the moves to be inserted simultaneously. The reference chart is identified with the terminal fixed-site chart before any slide is made. That identification imports no assertion about a moved marking or a face of the slide cube. \subsection{Protected sectors and completed charts} Fix \[ g\geq1,\qquad s\geq1,\qquad k\geq2s, \] and an injective assignment of ordered active pairs \begin{equation} a\longmapsto(i_a,j_a),\qquad a\in[s], \label{slide:eq:active-pair-assignment} \end{equation} such that the \(2s\) labels \(i_a,j_a\) are all distinct. On the outgoing boundary of the reference chart \(\mathfrak C_\varnothing\), write \(\bar m_t^\varnothing,\bar\ell_t^\varnothing\) for the geometric marked curves and \(m_t,\ell_t\subset Y\) for their images in the fixed abstract surface. These are the rows of the terminal attachment marking from \cref{common:sec-walker-data}; in particular, they realize \eqref{common:eq-terminal-homology-splitting} and \eqref{common:eq-terminal-lagrangian}. Their one-holed-torus support collars are pairwise disjoint and are disjoint from the old-region marking. For each \(a\in[s]\), choose a protected genus-two sector \(U_a\subset W\) containing exactly the complete marking data at the labels \(i_a,j_a\). At a vertex \(A\subseteq[s]\), the actual co-core at label \(t\) is denoted \(D_t^A\). The reference sector contains \(D_{i_a}^\varnothing,D_{j_a}^\varnothing\), the prescribed positive push-off \(D_{j_a}^{\mathrm{po}}\) of the second co-core, the positive product-zero-framed slide band, and the dual-longitude band. It also contains the complete old and slid cut-cellulations, all site collars, the outer detector collar with its framed crossing arc, the dual ribbons, and all signed faces and face pairings. The sector meets the closure of its complement in a single interface disk \(E_a\), supplied with a full product collar. The closures of the \(U_a\), including every named collar and protected product neighborhood, are pairwise disjoint. Moreover, \(U_a\) is disjoint from the feet, bands, co-cores, marked curves, framing collars, and product neighborhoods of every label outside \(\{i_a,j_a\}\). Labels in no active pair are called \emph{unused}. Thus, when direction \(a\) is inserted, the incoming boundary and its collar, the old genus-\(g\) region, every \(U_b\) with \(b\neq a\), and every unused handle together with a neighborhood of its complete marking lie in the identity exterior of \(U_a\). This is a pointwise separation hypothesis, not merely a disjointness assertion on homology classes. For \(A\subseteq[s]\), define \(\mathfrak C_A\) directly: in \(U_b\), use the complete old marked row if \(b\notin A\) and the complete positively slid row if \(b\in A\); on the common complement, use the old row. The protected closures are disjoint, so these insertions are simultaneous. This definition never passes through a history of partially attached or partially slid handles. Let \(\bar m_t^A,\bar\ell_t^A\) be the actual geometric rows on \(\partial_+(W,\mathfrak C_A)\), with their orientations, product framings, and support collars, and let \(D_t^A\) be the actual co-core disk in that completed marking. The abstract rows \(m_t^A,\ell_t^A\subset Y\) are obtained by inserting the corresponding complete coordinate row in every protected sector. The outgoing parameterization is the simultaneous gluing \begin{equation} p_A\colon\partial_+(W,\mathfrak C_A)\longrightarrow Y, \qquad p_A(\bar m_t^A)=m_t^A,\quad p_A(\bar\ell_t^A)=\ell_t^A. \label{slide:eq:completed-vertex-parameterization} \end{equation} The equalities retain labels, orientations, product framings, and support collars. Old and slid local rows agree literally on every full interface collar and on the common complement, so there is no independent postcomposition of one local piece of \(p_A\). The transported support collars remain pairwise disjoint: the collars within one active sector belong to its complete local model, distinct protected sectors have disjoint closures, and every unused collar lies in the common identity exterior. The declared vertex Lagrangian is the subspace \(\Lambda_A\) defined in \eqref{slide:eq:vertex-lagrangian}; its equality with \(\Lambda\) away from the reference vertex is not assumed here. \subsection{The terminal/reference-vertex interface} \label{slide:subsec:fixed-site-input} The same terminal compression body carries two bookkeeping systems. Fixed-site notation remembers the attachment history and ends at the state \(I=[k]\); slide notation starts with all handles present and calls its old completed marking \(A=\varnothing\). The following proposition is the entire interface between those systems. \begin{proposition}[Terminal/reference-vertex interface] \label{slide:prop:reference-vertex-interface} At the reference slide vertex, the completely marked compression body \((W,\mathfrak C_\varnothing)\colon X\to Y\) is the terminal simultaneous fixed-site trace. In particular, \begin{equation} m_t^\varnothing=m_t,\qquad \ell_t^\varnothing=\ell_t,\qquad \Lambda_\varnothing=\Lambda=\lambda_{[k]}. \label{slide:eq:reference-marking-interface} \end{equation} The equality of compression bodies retains the endpoint parameterizations, oriented co-cores, framed rows, signed cut faces and pairings, collars, ribbons, detector germs, and label data. The first two displayed equalities are equalities of embedded oriented product-framed curves in the fixed surface \(Y\). Let \begin{equation} \sigma_t^{\mathrm{ref}} :=\sigma_{t\mid[k]\setminus\{t\}} =T_{m_t}T_{\ell_t}T_{m_t}, \qquad \sigma_{t;\varnothing} :=T_{m_t^\varnothing}T_{\ell_t^\varnothing}T_{m_t^\varnothing}, \label{slide:eq:old-vertex-phase-word} \end{equation} where each word is extended by the identity outside its specified one-holed-torus collar. Then \(\sigma_{t;\varnothing}=\sigma_t^{\mathrm{ref}}\) as a supported marked mapping class. If \(t_1,\ldots,t_r\) are the labels for which \(\epsilon_t=1\), in any order, set \begin{equation} \sigma_{\mathrm{ref}}^\epsilon :=\sigma_{t_r}^{\mathrm{ref}}\cdots\sigma_{t_1}^{\mathrm{ref}}. \label{slide:eq:reference-phase-product} \end{equation} This mapping class and the associated Walker cylinder are independent of the displayed order. Define \(\sigma_{\mathrm{ref}}^\eta\) by the same rule using the labels for which \(\eta_t=1\). Define the directly completed phase-zero reference traces by giving \((W,\mathfrak C_\varnothing)\) and its literal reverse the weights \(c_\Sigma\) and \(r_\Sigma\), respectively. With the slide-state subscript \(\varnothing\), they satisfy the literal equalities \begin{align} J_\varnothing^0(\mathbf c) &=J_{[k]}^0(\mathbf c) =J_{\mathrm{sim}}^0(\mathbf c) \colon X\longrightarrow Y, \label{slide:eq:reference-phasezero-forward}\\ R_\varnothing^0(\mathbf r) &=R_{[k]}^0(\mathbf r) =R_{\mathrm{sim}}^0(\mathbf r) \colon Y\longrightarrow X. \label{slide:eq:reference-phasezero-reverse} \end{align} For arbitrary phase vectors, put \begin{align} J_\varnothing^\epsilon(\mathbf c,\mathbf u^+) &:= C\!\left(\sigma_{\mathrm{ref}}^\epsilon, U^+(\epsilon)\right)J_\varnothing^0(\mathbf c), \label{slide:eq:old-forward-completed-trace}\\ R_\varnothing^\eta(\mathbf r,\mathbf u^-) &:= R_\varnothing^0(\mathbf r) C\!\left(\sigma_{\mathrm{ref}}^\eta, U^-(\eta)\right). \label{slide:eq:old-reverse-completed-trace} \end{align} Then \begin{align} J_\varnothing^\epsilon(\mathbf c,\mathbf u^+) &=J_{[k]}^\epsilon(\mathbf c,\mathbf u^+) =J_{\mathrm{sim}}^\epsilon(\mathbf c,\mathbf u^+), \label{slide:eq:reference-full-forward-interface}\\ R_\varnothing^\eta(\mathbf r,\mathbf u^-) &=R_{[k]}^\eta(\mathbf r,\mathbf u^-) =R_{\mathrm{sim}}^\eta(\mathbf r,\mathbf u^-). \label{slide:eq:reference-full-reverse-interface} \end{align} The respective weights are \(c_\Sigma+U^+(\epsilon)\) and \(r_\Sigma+U^-(\eta)\). \end{proposition} \begin{proof} The interface is proved by comparing the two complete gluings piece by piece: the reference chart uses exactly the terminal old rows on every protected sector and on the common complement. Once those rows, collars, and endpoint parameterizations are identified, the phase words have the same supported curves as the terminal fixed-site words. Disjoint supports then make the phase factors commute and remove the intermediate Walker seams. The proposition thus transfers the terminal trace at the old vertex only; slide edges require the local insertion constructed below. The reference chart uses the complete old row in every protected sector and the old row on the common complement. Those are precisely the pieces of the terminal attachment marking, with the same product coordinates on every interface collar. Their simultaneous gluing is therefore the terminal parameterization, not a parameterization related to it by an unrecorded postcomposition. This proves the equality of compression bodies with complete marking and all the identities in \eqref{slide:eq:reference-marking-interface}. The last identity also follows directly by substituting \(m_t^\varnothing=m_t\) into \eqref{slide:eq:vertex-lagrangian} and comparing with \eqref{common:eq-terminal-lagrangian}. The same equality of framed rows and support collars gives \(\sigma_{t;\varnothing}=\sigma_t^{\mathrm{ref}}\) in \eqref{slide:eq:old-vertex-phase-word}. Distinct phase words have disjoint supports. The calculation in \cref{common:lem-disjoint-phase-cylinders} therefore shows both that their mapping classes commute and that every Walker correction between them vanishes. Hence \eqref{slide:eq:reference-phase-product} is independent of the chosen order and its labelled cylinders combine to weight \(U^+(\epsilon)\), or to \(U^-(\eta)\) in the reverse phase product. At phase zero, the directly completed reference cobordism and the fixed-site simultaneous trace have the same underlying oriented compression body (respectively its literal reverse), the same complete endpoint marking, and the same respective total weights \(c_\Sigma\) and \(r_\Sigma\). Thus they are literally the same Walker morphisms, proving \eqref{slide:eq:reference-phasezero-forward} and \eqref{slide:eq:reference-phasezero-reverse}. The simultaneous-trace clause of \cref{fixed:thm-main-package} identifies these morphisms with every fixed-site ordered history. Adding the identical supported phase words, with the zero corrections just calculated, proves \eqref{slide:eq:reference-full-forward-interface} and \eqref{slide:eq:reference-full-reverse-interface}, including the stated weights. \end{proof} The proposition transfers only the reference marking, the terminal fixed-site phase words, the label-attached weights, and the completed terminal traces. It transfers no word at \(A\neq\varnothing\), no slide edge or inverse, no edge conjugacy, no Walker seam across an edge, no two-face, and no finite slide-order conclusion. \subsection{Local-model identification and supported insertion} \label{slide:subsec-model-identification} \label{slide:subsec:local-model-identification} We now specify the geometric input for one positive edge. For every \(a\in[s]\), fix orientation-preserving product coordinate identifications that preserve the complete marking, \begin{align} \chi_a^-\colon (U_a,E_a,\text{old local data}) &\longrightarrow (U_\gamma,E_\gamma,\text{old local data}), \label{slide:eq:source-local-chart}\\ \chi_a^+\colon (U_a,E_a,\text{slid local data}) &\longrightarrow (U_\gamma^\gamma,E_\gamma,\text{slid local data}). \label{slide:eq:target-local-chart} \end{align} They use the same product coordinates on \(E_a\) and its full collar. Under this dictionary, the local labels \(1,2\) correspond respectively to \(i_a,j_a\), and \(D_{j_a}^{\mathrm{po}}\) corresponds to the local \(D_2^{\mathrm{po}}\). The positive band, dual band, cut balls, complete cellulation, signed faces and face pairings, co-core orientations, dual ribbons, detector, product framings, and every collar germ also agree with their local-model counterparts. \begin{lemma}[Local-model identification and identity extension] \label{slide:lem:local-model-identification} Let \(F_\gamma\colon U_\gamma\to U_\gamma^\gamma\) be the complete-marked positive-slide representative constructed in \cite[Theorem~2.2 and Appendix~F]{CompleteMarkedPentagons2026}. Then \begin{equation} f_a:=(\chi_a^+)^{-1}F_\gamma\chi_a^- \colon U_a\longrightarrow U_a \label{slide:eq:transported-local-map} \end{equation} has the coordinate-identity product germ on the full collar of \(E_a\). The piecewise formula \begin{equation} \widehat F_a(x)= \begin{cases} f_a(x),&x\in\operatorname{int}(U_a),\\ x,&x\in W\setminus\operatorname{int}(U_a) \end{cases} \label{slide:eq:supported-ambient-map} \end{equation} defines a smooth orientation-preserving diffeomorphism of \(W\), with a smooth inverse, supported in \(U_a\). It is the identity on the incoming boundary and its collar, the old genus-\(g\) region, every \(U_b\) with \(b\neq a\), every unused handle, and a neighborhood of all corresponding complete marking data. \end{lemma} \begin{proof} The local model supplies a map preserving the complete marking, not merely an underlying diffeomorphism. Transport through the two collar-preserving charts gives a map with the identity germ needed for the piecewise formula to glue smoothly. The support-separation hypotheses then extend that local map to all of \(W\) while fixing every other marked sector, so the same supported map can be reused at each slide state; only the typed boundary transition changes with \(A\). The cited Appendix~F constructs both \(F_\gamma\) and its inverse with the coordinate-identity germ on a full interface collar, while preserving every item in the preceding dictionary. Transport through \eqref{slide:eq:source-local-chart}-- \eqref{slide:eq:target-local-chart} preserves that germ. Consequently the two branches in \eqref{slide:eq:supported-ambient-map}, and separately the two inverse branches, agree on an open product neighborhood with all derivatives. They glue to mutually inverse smooth maps, and both branches have positive orientation sign. The pointwise fixation statement follows from the separation hypotheses: the incoming boundary and collar, the old region, every other protected sector, and every unused site lie, together with neighborhoods of their complete markings, in the identity exterior. \end{proof} For \(A\subseteq[s]\) and \(a\notin A\), use the same ambient map at every source vertex and define \begin{equation} F_{a\mid A}:=\widehat F_a \colon(W,\mathfrak C_A)\longrightarrow (W,\mathfrak C_{A\cup\{a\}}). \label{slide:eq:completed-edge-map} \end{equation} Inside \(U_a\), complete local transport carries the old row to the slid row; outside \(U_a\), every row is fixed pointwise. Hence \eqref{slide:eq:completed-edge-map} has the displayed source and target types. Its formula is independent of \(A\), since all already-slid sectors are disjoint from its support. Define the outgoing transition and its weight-zero Walker lift by \begin{equation} h_{a\mid A} :=p_{A\cup\{a\}}\circ (F_{a\mid A}|_{\partial_+W})\circ p_A^{-1}, \qquad H_{a\mid A}:=C(h_{a\mid A},0). \label{slide:eq:edge-transition-cylinder} \end{equation} Both endpoint parameterizations are part of this type. The zero is part of the definition; a cylinder of another weight is not a slide edge here. The transport also retains the residual-factor tests attached to the chosen local representative. A meridian-twist postfactor would change the winding of a transported dual ribbon, while the prescribed ribbon row has winding zero. A separating twist about the outer collar would change the full turn number of the framed detector crossing, while the transported detector has turn zero. After cutting along the two active co-cores, any remaining cell-fixing postfactor fixes every vertex and edge of the complete cellulation and the boundary of each disk or rectangular two-cell. Cellwise Alexander isotopies, paired across the signed faces, make that postfactor topologically isotopic to the identity relative to the complete cellular marking. This topological statement does not assert smooth-relative uniqueness. The smooth edge used here is the particular representative transported in \eqref{slide:eq:transported-local-map}. Its calibration uses the ribbons, detector, collars, and complete cellulation; it is not inferred from the four homology formulas proved later. The citation in \cref{slide:lem:local-model-identification} supplies one local geometric representative in a complete genus-two model, with its residual-factor calibration. Transport through the two product-coordinate identifications retains its identity on the full interface collar; the separation and identity-exterior hypotheses then give the spectator fixation proved above. These are the inputs to the completed edge. Walker equality, phase and weight identities, insertion for arbitrary spectators, and compatibility between slides are not supplied by that citation. The homology action, common Lagrangian, edge covariance, squares, and finite closure are established in the subsequent sections from the chosen edge.