\section{Compression bodies and supplied protected sectors} \label{sec:protected-sectors} Before a route can be compared, the local geometry on which it acts must be part of the input. This section identifies the completed compression body, the protected three-handle sector, and the precise meaning of “supplied.” The theorem is local in a precise sense. Its input is not an arbitrary completed compression body in which a suitable three-handle configuration must somehow be found. Rather, the configuration, all of its product coordinates, and the room needed for the subsequent constructions are supplied. This distinction is the first scope boundary of the result. \subsection{The completed compression body} Fix integers \(g\geq 1\) and \(N\geq 3\). We write \begin{equation} W:X\longrightarrow Y_{\mathrm{geom}} \label{eq:completed-compression-body} \end{equation} for a compact oriented compression body obtained from \(X\times[0,1]\), where \(X\) is a closed oriented surface of genus \(g\), by attaching \(N\) ordered product \(1\)-handles to the top. The arrow in \eqref{eq:completed-compression-body} records the incoming and outgoing boundaries; it is not a map of surfaces. The outgoing boundary is denoted \(\partial_+W=Y_{\mathrm{geom}}\). A \emph{completed row} consists of an ordered complete co-core system together with the data that will be held fixed under relative diffeomorphisms: signed cut-face charts, meridian and product-framed dual-longitude germs, ribbon and spoke germs, detector arcs, inward and two-sided normal collars, and an outgoing parameterization. Choose three pairwise distinct indices \[ i,j,k\in\{1,\ldots,N\}. \] The \emph{exterior completed row} \(\mathfrak C_{\mathrm{ext}}\) contains all old data and the co-cores \(D_t\), with all their marking data, for \(t\notin\{i,j,k\}\). It also includes a fixed identification of the old homology summand and a rational disk-kernel Lagrangian \begin{equation} \lambda_g\subset H_1(X;\Q). \label{eq:old-lagrangian} \end{equation} Every construction below is pointwise the identity on this exterior row. When \(N=3\), the collection of unused co-cores is empty; all statements involving it are then read as empty sums or vacuous families. \subsection{The protected three-handle sector} \begin{definition}[Supplied protected sector] \label{def:supplied-protected-sector} A fully protected standard sector \(\cL_{ijk}\) consists of the following mutually compatible smooth, oriented, product-parameterized data. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=2.7em] \item A top product collar \begin{equation} \chi:X\times[1-\tau,1]\hookrightarrow W \label{eq:top-product-collar} \end{equation} and a closed oriented disk \(Q\subset X\times\{1\}\), equipped with a fixed disk chart and a common normal-height coordinate. \item Six mutually disjoint attaching feet \(q_r^\pm\Subset\Int Q\), \(r\in\{i,j,k\}\), and three oriented, product-framed handle embeddings \begin{equation} \iota_r:[-2,2]\times D^2\hookrightarrow W,\qquad D_r=\iota_r(\{0\}\times D^2). \label{eq:three-handle-charts} \end{equation} The positive normal to \(D_r\) is \(d\iota_r(\partial_x)\). The input contains full co-core collars and fixed meridian and dual-longitude product germs. \item Separated height slabs, a number \(\varepsilon>0\), and the mutually disjoint product push-offs \begin{equation} i_0;\qquad j_0,j_\varepsilon;\qquad k_0,k_\varepsilon,k_{2\varepsilon}. \label{eq:protected-push-offs} \end{equation} Every push-off carries its orientation, positive normal, parameterized full collar, and product-zero framing. A nonzero subscript records an auxiliary parallel copy, not an additional disk in a completed row. \item Three oriented product-zero-framed rectangular band germs, with their full collars and smoothing boxes: \begin{equation} \begin{array}{c|c|c} \text{band}&\text{moving attachment}&\text{positive carrier attachment}\\ \hline \beta_0&j_0&k_\varepsilon\\ \alpha&i_0&j_\varepsilon\\ \beta_1&j_\varepsilon&k_{2\varepsilon}. \end{array} \label{tab:three-supplied-bands} \end{equation} Each band germ is supplied by a rectangular \(\bar\gamma\) in the high boundary whose interior is pushed slightly inward through the frozen high-boundary product collar, with its attaching short sides fixed. Its product-zero framing is the transported constant normal of that rectangle. Thus these are boundary-parallel product bands in the specified charts, not arbitrary embedded framed rectangles. There is one product-normal translation \(\nu_\varepsilon\), defined on the complete relevant band and smoothing germs, such that \begin{equation} \nu_\varepsilon(j_0)=j_\varepsilon,\qquad \nu_\varepsilon(k_\varepsilon)=k_{2\varepsilon},\qquad \nu_\varepsilon(\beta_0)=\beta_1 \label{eq:exact-band-translation} \end{equation} to every order in the product coordinates. Thus \(\beta_1\) is the literal normal translate of \(\beta_0\), not merely an independently chosen parallel band. \item One simultaneous oriented corner-rounding rule for the four descendant disks \begin{align} J&=j_0\#_{\beta_0}k_\varepsilon,\notag\\ J^+&=j_\varepsilon\#_{\beta_1}k_{2\varepsilon},\notag\\ I_A&=i_0\#_\alpha j_\varepsilon,\notag\\ I_\star&=\operatorname{smooth} (i_0-\alpha-j_\varepsilon-\beta_1-k_{2\varepsilon})\notag\\ &=i_0\#_\alpha J^+ =(I_A)\#_{\beta_1}k_{2\varepsilon}. \label{eq:four-descendant-disks} \end{align} The two descriptions of \(I_\star\) specify the same embedded disk with the same full product germ. Moreover \(J^+=\nu_\varepsilon(J)\) with its entire product germ. Retain a closed subdisk \(\Omega_j\Subset I_A\), with a full collar, in the realized \(j_\varepsilon\)-lobe. The fixed parameterizations of \(I_A\) and \(I_\star\) agree on \(\Omega_j\); every face sector, ribbon attachment, and spoke germ belonging to that lobe lies there away from the \(\beta_1\)-attachment. Thus the \(AAC\) moving-face map carries this attached \(j\)-portion and its complete marking rather than treating it as an omitted spectator. \item The rounded active sector \begin{equation} U_{ijk}=\operatorname{Round}\left( \chi(Q\times[1-\tau,1])\cup \bigcup_{r\in\{i,j,k\}}\iota_r([-2,2]\times D^2) \right), \label{eq:rounded-active-sector} \end{equation} its unique exterior interface disk \(E\), and a fixed bicollar \(E\times[-\rho,\rho]\). The symbol \(E\) will always denote this global interface. The edge-local interface disks introduced in \cref{sec:endpoint-atlases} are denoted \(E_e\) and are different objects. \item The exterior attachment \((\mathfrak C_{\mathrm{ext}},\lambda_g)\) and one common exterior parameterization to a fixed abstract outgoing surface \(Y\). It is literally product-coordinate identity on the old region and all unused handles, and its support is disjoint from the active data above. \item Reserve neighborhoods and positive separation constants for the seven cut-face sites, six branch corridors, one detector corridor, the five possible internal-interface sites, and a complementary-disk annulus with an inward collar. These are capacity and clearance data. They do not themselves constitute the faces, ribbons, pairings, atlases, or amalgamations constructed later. \end{enumerate} \end{definition} \begin{figure}[htbp] \centering \begin{tikzpicture}[ region/.style={draw,rounded corners=5pt,minimum height=3.0cm}, handle/.style={draw,circle,minimum size=7mm,inner sep=0pt}, label/.style={font=\small,align=center} ] \node[region,minimum width=9.6cm] (W) {}; \node[anchor=north west] at ($(W.north west)+(.15,-.15)$) {\(W:X\to Y_{\mathrm{geom}}\)}; \node[region,minimum width=4.8cm,minimum height=1.65cm] (U) at (1.45,0) {}; \node[anchor=north west,font=\small] at ($(U.north west)+(.12,-.12)$) {\(U_{ijk}\)}; \node[handle] (i) at (.25,-.15) {\(i\)}; \node[handle] (j) at (1.45,-.15) {\(j\)}; \node[handle] (k) at (2.65,-.15) {\(k\)}; \draw[very thick] ($(U.west)+(0,-.55)$) -- ($(U.west)+(0,.55)$); \node[label,anchor=east] at ($(U.west)+(-.1,0)$) {exterior completed row\\\(\mathfrak C_{\mathrm{ext}}\)}; \node[label,anchor=south] at ($(U.north)+(0,.13)$) {interface disk \(E\) with full bicollar}; \node[label] at (1.45,-1.85) {three supplied handle charts, product collars, bands, and reserve regions}; \end{tikzpicture} \caption{Schematic incidence data for the supplied protected sector. The diagram records the active three-handle region, the exterior row, and their single collared interface. It does not prescribe an embedding, page orientation, band placement, framing arrow, or collar width beyond \cref{def:supplied-protected-sector}.} \label{fig:protected-sector-schematic} \end{figure} The disjointness in this definition includes the interiors, complete collars, and smoothing neighborhoods of the three bands, except at their displayed attachments on distinct parallel disk copies. In particular, \begin{equation} j_0\cap j_\varepsilon=\varnothing,\qquad k_0\cap k_\varepsilon =k_0\cap k_{2\varepsilon} =k_\varepsilon\cap k_{2\varepsilon}=\varnothing, \label{eq:height-separation} \end{equation} and \begin{equation} \alpha\cap(\beta_0\cup\beta_1)=\varnothing,\qquad \beta_0\cap\beta_1=\varnothing \label{eq:band-separation} \end{equation} away from those prescribed attachments. These geometric separations, not identities in homology, are what make the five disk rows below embedded. \subsection{The local positive two-site theorem used as input} The five-edge argument does not manufacture the elementary slide class from a homology matrix. It uses the following local theorem, stated here in the exact form consumed below and proved in \cref{app:local-positive-slide}. \begin{theorem}[Positive two-site slide with complete markings] \label{thm:positive-two-site-input} Let \(D_1,D_2\) be disjoint oriented co-core disks in a frozen oriented two-handle sector. Write \(m_r=\partial D_r\), and equip this \emph{old} ordered disk system with a complete local marking: signed disk-face charts with full normal collars, actual embedded meridian and product-framed dual-longitude rows, ribbon and detector germs, and one fixed exterior interface disk \(E_\gamma\) with its product neighborhood. The old dual longitudes \(\ell_1,\ell_2\) meet the meridians geometrically by \begin{equation} \begin{aligned} m_r\pitchfork\ell_r&=\{x_r\}\text{ with positive sign},\\ m_1\cap m_2&=\ell_1\cap\ell_2=\varnothing,\qquad m_r\cap\ell_s=\varnothing\quad(r\ne s). \end{aligned} \label{eq:local-old-geometric-row} \end{equation} In particular, \(m_r\mathbin{\cdot}\ell_s=\delta_{rs}\). Fix a constant-height positive product push-off in the supplied co-core collar, \(D_2^{\mathrm{po}}\) with \[ D_2^{\mathrm{po}}\cap(D_1\cup D_2)=\varnothing, \] a positive product-zero-framed band \(\gamma\) from \(D_1\) to \(D_2^{\mathrm{po}}\), supplied as a high-boundary rectangle \(\bar\gamma\) whose interior is pushed slightly inward through the frozen product collar while its attaching short sides remain fixed, and the slid disk system \[ D_1^\gamma=D_1\natural_\gamma D_2^{\mathrm{po}},\qquad D_2^\gamma=D_2. \] The attaching rectangles miss \(x_1,x_2\). The band interior and its smoothing box are disjoint from the two co-cores and every old protected marking except at the prescribed attaching rectangles. Fix the prescribed embedded product-zero-framed dual band \(\gamma^\vee\): it is the narrow parallel rectangle routed on the specified side of the boundary trace of \(\gamma\), from the \(\ell_2\)-side to a positive push-off of the \(\ell_1\)-side, with its second short side oriented oppositely to \(\ell_1\). Its interior misses the four old curves except at those prescribed ends. Set geometrically \begin{equation} m_1^\gamma=\partial D_1^\gamma,\qquad m_2^\gamma=m_2,\qquad \ell_1^\gamma=\ell_1,\qquad \ell_2^\gamma =\ell_2\natural_{\gamma^\vee}(-\ell_1)^+ . \label{eq:local-geometric-slide-rows} \end{equation} Assume that the fixed exterior interface \(E_\gamma\) cuts off a closed product-ended two-handle subsector containing the two co-cores, this push-off, both prescribed bands, their smoothing boxes, and all local protected marking germs. Its product neighborhood is disjoint from those germs. Thus \(E_\gamma\) is part of the supplied relative boundary, not a support disk selected by the proof. Then the slid ordered disk system admits a complete local marking whose actual meridian and product-framed dual-longitude rows are precisely those in \eqref{eq:local-geometric-slide-rows}, while retaining the fixed exterior-interface germ. The slid disk system is complete in the two-handle sector. Cutting the two completely marked systems gives marked balls \[ B_-^{\mathrm{LS}},\qquad B_+^{\mathrm{LS}}. \] Moreover, there is an orientation-preserving diffeomorphism \[ \phi_\gamma^{\mathrm{cut}}: B_-^{\mathrm{LS}}\longrightarrow B_+^{\mathrm{LS}} \] with the following properties. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=2.7em] \item Its boundary map carries the complete old cut-cellulation to the complete slid cut-cellulation. It transports the actual embedded meridian and dual-longitude rows, every marked ribbon and detector, and is the coordinate identity on \(E_\gamma\) and its product germ. \item For each of the two cut disks it intertwines the minus-to-plus pairing and its inverse on a common full-normal collar. In the paired coordinates the normal variable is preserved, so the two descended expressions agree with all jets after regluing. \item On a full outer collar it has the prescribed product formula. Its relative class is fixed as follows. The complete cellwise boundary reference, denoted \(B0\), is first smoothed without changing any protected germ. A collar-product radial ball extension is then smoothed relative to that full collar. The resulting diffeomorphism lies in the topological relative class \(B5\) of the radial extension: if \(p,q\) are the two inner-ball maps with the same boundary trace, their quotient fixes the inner sphere and the explicit Alexander family \begin{equation} H_s(x)= \begin{cases} s\,q^{-1}p(x/s),&|x|\le s,\\ x,&s\le |x|\le1, \end{cases} \qquad 0