\section{Main results} \label{common:sec-main-results} This paper studies two operations on marked handles. First, what is the composite of forward and reverse attachments when several marked sites are present? Second, what extra work is needed when the marking itself changes along a disjoint handleslide cube with complete markings? We call the composite of forward and reverse attachments a \textit{mixed double}, allowing their phase labels to be chosen independently. The first question is local and fixed-site; the second concerns the vertices of the cube and requires supplied protected slide sectors. Classical cut-system and Heegaard-diagram complexes already contain square and pentagon relation cells for handleslides \cite{HatcherThurston1980,JuhaszThurstonZemke2021}. The claims here are therefore limited to the printed refinement: supplied pairwise-disjoint protected sectors, complete endpoint markings, actual supported maps, a common Walker Lagrangian, and literal equalities with the declared weights. No priority claim is made for a square or order relation after those qualifiers are dropped. In the case without distinguished disks, Qin's sutured compression-body complex likewise uses isotopy classes of cut-systems and triangle, square, and pentagon relations to prove simple connectivity \cite[Theorem~1.1, Definition~2.5, and Section~4, especially Definition~4.2]{Qin2025}. Our disjoint cube instead keeps the specified complete marking and selected smooth representatives; it does not supply the full move-complex conclusion. At fixed sites, the Boolean coordinate records which handles have been attached, and different orders are histories with common endpoints in one pulled-back chart. In the slide layer, every Boolean coordinate records a completed change of marking; the source and target rows of later edges therefore depend on the current vertex. Commutation on homology alone does not identify the corresponding Walker morphisms, because endpoint Lagrangians and ordered Maslov corrections must also agree. Our fixed-site calculation begins locally. We retain the actual supported half-turn and separating-disk twist, calculate the four weighted same-site composites, and prove every heterogeneous interchange before passing to order-independent histories. An independently constructed simultaneous trace then identifies the topology of the mixed double without retaining an artificial decomposition in the output. This yields \cref{fixed:thm-main-package}. \begin{headlinetheorem}[Fixed-site phase double] \label{fixed:thm-conceptual-double} A matching forward/reverse phase leaves one local \(S^1\times S^2\) factor, while a mismatching phase leaves an identity cylinder; the global mixed double is the product of these local contributions, with the Walker weight carried separately by the identity-cylinder part. \end{headlinetheorem} For the moving layer, one fully specified two-site positive-slide representative is imported from the companion pentagon theorem \cite[Theorem~2.2 and Appendix~F]{CompleteMarkedPentagons2026}. Here a protected sector is a specified genus-two neighborhood containing the complete labelled local data for one slide direction and meeting its complement along a collared interface disk. The complete marking includes the oriented curves, framings, collars, and cut-face data needed by the local representative. We prove the additional statements needed to place disjoint copies into those sectors: compatibility of all completed vertex rows, actual supported edge maps and inverses, signed integral homology, a common rational Lagrangian, phase-zero and all-phase Walker covariance, literal two-faces, and finite-route closure. These arguments produce \cref{slide:thm-complete-marked-disjoint-slide-cube}; none is attributed to the imported two-site construction. \begin{headlinetheorem}[Disjoint handleslide cube] \label{slide:thm-conceptual-cube} A cube edge is not merely a commuting homology matrix: it is a completed marked Walker cobordism whose endpoint Lagrangians, phase data, and weights have been matched. Once those data are aligned, every two-face is literally equal, and every finite route has the same composite. \end{headlinetheorem} \subsection{Fixed marked sites} \label{fixed:subsec-main-result} For a positive integer \(q\), write \([q]=\{1,\ldots,q\}\). Fix \(g,k\geq 1\), a connected marked outgoing surface \((\Sigma_g^{\mathrm{out}},\lambda_g)\), and \(k\) pairwise-disjoint, distinctly labelled disk sites with disjoint closures and complete local markings. Fix phase vectors \(\epsilon,\eta\in\{0,1\}^k\), label-attached weight vectors \[ \mathbf c,\mathbf r,\mathbf u^+,\mathbf u^-\in\mathbb Z^k, \] and independent attachment orders \(\alpha,\beta\in S_k\). A coordinate \(u_t^+\) with \(\epsilon_t=0\), or \(u_t^-\) with \(\eta_t=0\), is inactive and ignored. Write \[ n_t=c_t+r_t+\epsilon_tu_t^++\eta_tu_t^-, \qquad N=\sum_{t=1}^k n_t, \] and define the Hamming distance by \[ d_H(\epsilon,\eta) :=\lvert\{t\in[k]:\epsilon_t\neq\eta_t\}\rvert . \] Then \[ \mathcal M(\epsilon,\eta) =\{t\in[k]:\epsilon_t=\eta_t\}, \qquad b=\lvert\mathcal M(\epsilon,\eta)\rvert =k-d_H(\epsilon,\eta). \] For \(I\subseteq[k]\), let \(\Sigma_I^{\mathrm{out}}\) be the marked outgoing surface obtained by attaching exactly the sites in \(I\). For \(t\notin I\), the constructed phase arrows have types \[ J_{t\mid I}^{e}:\Sigma_I^{\mathrm{out}} \longrightarrow\Sigma_{I\cup\{t\}}^{\mathrm{out}}, \qquad R_{t\mid I}^{v}:\Sigma_{I\cup\{t\}}^{\mathrm{out}} \longrightarrow\Sigma_I^{\mathrm{out}}, \qquad e,v\in\{0,1\}. \] For \(n\in\mathbb Z\), write \(C(\operatorname{id}_{\Sigma},n)\) for the identity mapping cylinder of Walker weight \(n\), and \(\mathcal D_{t\mid I}(n)\) for the preferred local double of weight \(n\) in \(\operatorname{End}(\Sigma_I^{\mathrm{out}})\). At preferred weight zero, write \(1=C(\operatorname{id}_{\Sigma_I^{\mathrm{out}}},0)\) and \(\mathcal D=\mathcal D_{t\mid I}(0)\) in that typed endomorphism set. The symbols \(J_\alpha^\epsilon,R_\beta^\eta\) denote the corresponding ordered traces, \(J_{\mathrm{sim}}^\epsilon,R_{\mathrm{sim}}^\eta\) their independently constructed simultaneous counterparts, and \(M_{\alpha,\beta}^{\epsilon,\eta}:=R_\beta^\eta J_\alpha^\epsilon\) their mixed double. Fixed weight arguments are suppressed when no ambiguity can arise, and \((-)^\dagger\) denotes source orientation reversal. The detailed constructions appear in \cref{fixed:sec-local-calculus,fixed:sec-order-traces,fixed:sec-match-dagger}. The weighted same-site table identifies the local effect of each phase pair in one pulled-back marked chart. The heterogeneous interchanges then let arbitrary histories be reordered and compared with an independently constructed simultaneous trace. In that geometric trace, the matching pieces determine the connected-sum topology, while the accumulated weight belongs to the identity cylinder. Source reversal is applied after this marked source topology has been identified. \paragraph{Smallest fixed-site picture.} When there is one active site, the four phase choices already display the local dichotomy: matching forward and reverse phases produce the preferred local double, while either mismatch produces an identity cylinder. This is the smallest setting in which the phase labels have geometric meaning. It also shows why the later multi-site result is not just a permutation of attachments: the local type is determined before histories are reordered, and the Walker weight is carried by the resulting cylinder. \begin{figure}[htbp] \centering \[ \begin{array}{c|cc} & \eta_t=0 & \eta_t=1\\ \hline \epsilon_t=0 & \mathcal D_{t\mid I}(n_t) & C(\operatorname{id}_{\Sigma_I^{\mathrm{out}}},n_t)\\ \epsilon_t=1 & C(\operatorname{id}_{\Sigma_I^{\mathrm{out}}},n_t) & \mathcal D_{t\mid I}(n_t) \end{array} \] \caption{The one-site phase diagram. Diagonal phase choices match and leave a local \(S^1\times S^2\) factor; off-diagonal choices leave the identity cylinder. The Walker weight \(n_t\) is shown separately from this topological dichotomy.} \label{fixed:fig-one-site-phase-diagram} \end{figure} \paragraph{A worked \(k=2\) phase choice.} Take \(\epsilon=(0,1)\) and \(\eta=(1,1)\). Site \(1\) is a mismatch and therefore contributes an identity cylinder; site \(2\) is a match and contributes one local double. Thus \[ d_H(\epsilon,\eta)=1,\qquad \mathcal M(\epsilon,\eta)=\{2\},\qquad b=1, \] and for arbitrary allowed weights the theorem gives \[ M^{\epsilon,\eta}\cong C(\operatorname{id}_{\Sigma_g^{\mathrm{out}}},N) \mathbin{\#}(S^1\times S^2,0), \qquad \beta_1(M^{\epsilon,\eta})=2g+1. \] This specialization exposes both local outcomes while leaving the order of the two attachments irrelevant. \begin{theorem}[Fixed-site phase choices and completed traces] \label{fixed:thm-main-package} Uniformly over the preceding data, the following statements hold. \begin{enumerate} \item The Boolean family of attachment states, its Walker Lagrangians, and every phase-zero or phase-one forward and reverse handle arrow are fully typed in one pulled-back Walker chart. The local weights are \(c_t+\epsilon_tu_t^+\) and \(r_t+\eta_tu_t^-\), respectively, and every spectator marking is fixed. \item For every \(t\in[k]\) and \(I\subseteq[k]\setminus\{t\}\), the localized marked half-turn, fourth-power separating-disk twist, capping-kernel identity, the two reverse-ab\-sorption identities, and the elementary source-reversal identities hold with their actual Walker seams. In particular, source reversal inverts the supported slope and negates its weight; it is not a Hilbert-space adjoint and is not a slide inverse. \item The weighted same-site table is \[ R_{t\mid I}^{\eta_t}J_{t\mid I}^{\epsilon_t} = \begin{cases} \mathcal D_{t\mid I}(n_t),&\epsilon_t=\eta_t,\\ C(\operatorname{id}_{\Sigma_I^{\mathrm{out}}},n_t), &\epsilon_t\neq\eta_t, \end{cases} \] for every \(t\in[k]\) and \(I\subseteq[k]\setminus\{t\}\). Moreover, for every \(n\in\mathbb Z\), \[ \mathcal D_{t\mid I}(n) \cong C(\operatorname{id}_{\Sigma_I^{\mathrm{out}}},n) \mathbin{\#}(S^1\times S^2,0). \] Hence the preferred-weight table is \((\mathcal D,1,1,\mathcal D)\), and a matching-phase entry is not an identity cylinder of another weight. \item All four forward and all four reverse phase interchanges hold, including the heterogeneous cases, with subset tags and label-attached weights unchanged. Each fixed phase vector therefore has literal order independence. The associated Coxeter-shaped diagrams are diagrams of equality only; they construct no comparison cylinder and no permutation action. \item For every \(\alpha,\beta\in S_k\), the ordered histories agree literally with independently constructed simultaneous traces: \[ J_\alpha^\epsilon(\mathbf c,\mathbf u^+) =J_{\mathrm{sim}}^\epsilon(\mathbf c,\mathbf u^+), \qquad R_\beta^\eta(\mathbf r,\mathbf u^-) =R_{\mathrm{sim}}^\eta(\mathbf r,\mathbf u^-). \] Their homology, transported Lagrangians, complete endpoint markings, and Walker weights all agree. \item The mixed double \(M_{\alpha,\beta}^{\epsilon,\eta}\) is independent of both attachment orders; denoting its common value by \(M^{\epsilon,\eta}\), it satisfies \[ M^{\epsilon,\eta} \cong C(\operatorname{id}_{\Sigma_g^{\mathrm{out}}},N) \mathbin{\#} \left( \mathop{\#}_{t\in\mathcal M(\epsilon,\eta)} (S^1\times S^2,0) \right), \qquad \beta_1(M^{\epsilon,\eta})=2g+b. \] It is the identity cylinder of weight \(N\) exactly when \(b=0\); for \(b>0\) it is not an identity cylinder of any weight. No separating spheres or embedded labelled match set are retained as output data. \item Source reversal is contravariant, reverses products, and negates every\linebreak[4] Walker weight. In the order notation above, \[ (J_\alpha^\epsilon)^\dagger =R_\alpha^\epsilon(-\mathbf c,-\mathbf u^+), \qquad (R_\beta^\eta)^\dagger =J_\beta^\eta(-\mathbf r,-\mathbf u^-), \] and \[ (M_{\alpha,\beta}^{\epsilon,\eta})^\dagger =M_{\beta,\alpha}^{\eta,\epsilon} (-\mathbf r,-\mathbf c,-\mathbf u^-,-\mathbf u^+). \] \end{enumerate} Here equality means literal equality in the indicated marked Walker Hom set, including its integer weight. The symbol \(\cong\) denotes a boundary-marking-preserving diffeomorphism retaining the displayed weight data. The theorem includes \(g=1\), \(k=1\), empty phase words, complementary phase vectors, and arbitrary positive, zero, or negative active weights. \end{theorem} \subsection{Disjoint positive-slide sectors} \label{slide:subsec-main-result} Now fix \(s\geq1\) and \(k\geq2s\), a connected compression body with complete marking \[ W\colon X=\Sigma_g^{\mathrm{out}}\longrightarrow Y=\Sigma_{g+k}^{\mathrm{out}}, \] and a rational disk-kernel Lagrangian \(\lambda_g\subset H_1(X;\mathbb Q)\). The old marking fixes an inclusion \(H_1(X;\mathbb Q)\hookrightarrow H_1(Y;\mathbb Q)\), through which we regard \(H_1(X;\mathbb Q)\) and \(\lambda_g\) as subspaces of \(H_1(Y;\mathbb Q)\). Supply an injective ordered-pair assignment \[ a\longmapsto(i_a,j_a),\qquad a\in[s], \] with \(2s\) distinct active labels, and \(s\) pairwise-disjoint protected genus-two sectors carrying the complete local data and positive product-zero-framed slide specified in \cref{slide:sec-setup}. At the reference chart, assume that the one-holed-torus phase-support collars of all \(k\) labels are pairwise disjoint. For \(A\subseteq[s]\), let \(\mathfrak C_A\) be the directly completed vertex. For \(a\notin A\), let \(F_{a\mid A}\) be the supported compression-body edge, \(h_{a\mid A}\) its outgoing transition, and \[ H_{a\mid A}=C(h_{a\mid A},0) \] its marked weight-zero Walker cylinder. In the complete old marking, declare \[ H_1(Y;\mathbb Q) =H_1(X;\mathbb Q)\oplus \bigoplus_{t=1}^k \operatorname{span}_{\mathbb Q}\{[m_t],[\ell_t]\}, \] and \[ \Lambda=\lambda_g\oplus \bigoplus_{t=1}^k\langle[m_t]\rangle_{\mathbb Q}. \] For later weight formulas, put \[ c_\Sigma=\sum_{t=1}^k c_t, \qquad r_\Sigma=\sum_{t=1}^k r_t, \qquad U^+(\epsilon)=\sum_{\epsilon_t=1}u_t^+, \qquad U^-(\eta)=\sum_{\eta_t=1}u_t^-. \] The slide proof first fixes the complete protected sectors and directly completed vertices. It then constructs the actual edge maps with common collars, computes their signed homology actions, and proves the common rational Lagrangian. Next it checks Walker covariance for every phase and weight. The final step compares disjoint edges pointwise, which yields literal squares, finite route independence, and telescoping. \begin{figure}[htbp] \centering \begin{tikzpicture}[ vertex/.style={draw,rounded corners=2pt,inner sep=3pt}, edge/.style={-{Latex[length=2mm]},thick}, support/.style={draw,rounded corners=5pt,minimum width=1.65cm,minimum height=2.2cm} ] \node[draw,minimum width=4.4cm,minimum height=3.4cm] (W) at (-3.55,0) {}; \node[anchor=north west] at ($(W.north west)+(.12,-.12)$) {\(W\)}; \node[support] (Ua) at (-4.45,0) {\(U_a\)}; \node[support] (Ub) at (-2.65,0) {\(U_b\)}; \draw[edge] ($(Ua.west)+(.2,-.35)$) -- node[below=2pt] {\(\widehat F_a\)} ($(Ua.east)+(-.2,-.35)$); \draw[edge] ($(Ub.west)+(.2,-.35)$) -- node[below=2pt] {\(\widehat F_b\)} ($(Ub.east)+(-.2,-.35)$); \node[anchor=north,align=center,font=\small,text width=5.7cm] at (-3.55,-1.95) {disjoint protected supports;\\identity on the common complement}; \node[vertex] (A) at (0.65,-.75) {\(A\)}; \node[vertex] (Aa) at (2.85,-.75) {\(A\cup\{a\}\)}; \node[vertex] (Ab) at (0.65,1.0) {\(A\cup\{b\}\)}; \node[vertex] (Aab) at (2.85,1.0) {\(A\cup\{a,b\}\)}; \draw[edge] (A) -- node[below,font=\scriptsize] {\(F_{a\mid A}\)} (Aa); \draw[edge] (A) -- node[left,font=\scriptsize] {\(F_{b\mid A}\)} (Ab); \draw[edge] (Aa) -- node[right,font=\scriptsize] {\(F_{b\mid A\cup\{a\}}\)} (Aab); \draw[edge] (Ab) -- node[above=5pt,font=\scriptsize] {\(F_{a\mid A\cup\{b\}}\)} (Aab); \node[align=center,font=\small,text width=3.5cm] at (1.75,-1.8) {the two-site Boolean \(2\)-cube}; \end{tikzpicture} \caption{The local-support picture and the smallest disjoint handleslide cube. The left panel records only disjoint supports and identity on their common complement; it encodes no unstated meridian, longitude, framing, or orientation data. In the square, the intermediate-set tags are part of every edge, and the two composites are equal as complete-marked smooth maps before passing to homology or Walker cylinders.} \label{slide:fig-two-site-square} \end{figure} \begin{theorem}[Disjoint positive handleslide cube with complete markings] \label{slide:thm-complete-marked-disjoint-slide-cube} Uniformly over all of the preceding slide data, the following six conclusions hold. \begin{enumerate} \item Every \(\mathfrak C_A\), \(F_{a\mid A}\), \(h_{a\mid A}\), and \(H_{a\mid A}\) is well defined with complete marking and cube-compatible endpoint rows. The edge \(F_{a\mid A}\) has a smooth inverse, has the identity germ on the full interface collar, and fixes pointwise the incoming boundary, the old region, every other protected sector, and every unused handle. \item Every edge fixes the old and spectator homology summands and has active integral action \begin{align*} (h_{a\mid A})_*[m_{i_a}^A] &=[m_{i_a}^{A\cup\{a\}}] =[m_{i_a}^A]+[m_{j_a}^A],& (h_{a\mid A})_*[\ell_{i_a}^A] &=[\ell_{i_a}^{A\cup\{a\}}]=[\ell_{i_a}^A],\\ (h_{a\mid A})_*[m_{j_a}^A] &=[m_{j_a}^{A\cup\{a\}}]=[m_{j_a}^A],& (h_{a\mid A})_*[\ell_{j_a}^A] &=[\ell_{j_a}^{A\cup\{a\}}] =[\ell_{j_a}^A]-[\ell_{i_a}^A]. \end{align*} This action is integral symplectic. Every vertex Lagrangian satisfies \(\Lambda_A=\Lambda\), and every edge preserves \(\Lambda\) in both directions. The actual inverse carries the target marked row back to the source marked row; it is not the separate operation of expressing source classes in target coordinates. \item For arbitrary \(\mathbf c,\mathbf r\), the forward and reverse ordinary phase-zero identities are separately typed, and their Walker lifts are \[ H_{a\mid A}J_A^0(\mathbf c) =J_{A\cup\{a\}}^0(\mathbf c), \qquad R_{A\cup\{a\}}^0(\mathbf r)H_{a\mid A} =R_A^0(\mathbf r). \] The two ordered Maslov corrections vanish independently. The common weights are \(c_\Sigma\) and \(r_\Sigma\), respectively, and the completed morphisms retain no intrinsic decomposition of these totals. \item At each completed vertex, every phase word is defined from its actual oriented and framed meridian--longitude row. The supported labelled words are Walker-exact and order independent, agree with the fixed-site words at \(A=\varnothing\), and satisfy complete-marking conjugacy across an edge. All four ordered phase-cylinder corrections vanish in their actual multiplication orders. Consequently \[ H_{a\mid A}J_A^\epsilon(\mathbf c,\mathbf u^+) =J_{A\cup\{a\}}^\epsilon(\mathbf c,\mathbf u^+), \] \[ R_{A\cup\{a\}}^\eta(\mathbf r,\mathbf u^-)H_{a\mid A} =R_A^\eta(\mathbf r,\mathbf u^-), \] with common weights \(c_\Sigma+U^+(\epsilon)\) and \(r_\Sigma+U^-(\eta)\). Weight coordinates remain attached to their labels during every replacement; only the displayed totals are intrinsic to the completed morphisms. \item For \(A\subseteq[s]\) and distinct \(a,b\notin A\), the two routes around the face first agree as literal smooth maps on \(W\): \begin{equation} F_{b\mid A\cup\{a\}}F_{a\mid A} =F_{a\mid A\cup\{b\}}F_{b\mid A}. \label{slide:eq-main-smooth-square} \end{equation} They therefore induce the same marked outgoing map. Their two Walker corrections vanish separately and give the literal weight-zero square \begin{equation} H_{b\mid A\cup\{a\}}H_{a\mid A} =H_{a\mid A\cup\{b\}}H_{b\mid A}. \label{slide:eq-main-walker-square} \end{equation} \item For a slide order \(\omega=(\omega_1,\ldots,\omega_s)\), put \(A_q^\omega=\{\omega_1,\ldots,\omega_q\}\), \(A_0^\omega=\varnothing\), and \[ H_\omega= H_{\omega_s\mid A_{s-1}^\omega}\cdots H_{\omega_1\mid\varnothing}. \] All \(H_\omega\) have one common value \(H_{[s]}\) of Walker weight zero. The involutive, distant, and braid-shaped adjacent-replacement diagrams close as equalities, but yield no retained swap morphism or permutation action. The endpoint covariances telescope to \begin{align*} H_{[s]}J_\varnothing^0(\mathbf c) &=J_{[s]}^0(\mathbf c),& R_{[s]}^0(\mathbf r)H_{[s]} &=R_\varnothing^0(\mathbf r),\\ H_{[s]}J_\varnothing^\epsilon(\mathbf c,\mathbf u^+) &=J_{[s]}^\epsilon(\mathbf c,\mathbf u^+),& R_{[s]}^\eta(\mathbf r,\mathbf u^-)H_{[s]} &=R_\varnothing^\eta(\mathbf r,\mathbf u^-). \end{align*} \end{enumerate} \end{theorem} The terminal subscript \([s]\) in the last clause is the vertex at which all \(s\) slide directions have been applied; it is not the handle-label set \([k]\). Theorem~\ref{slide:thm-complete-marked-disjoint-slide-cube} includes \(g=1\), \(s=1\), \(k=2s\), empty phase words, arbitrary zero or negative weight coordinates, the reference and terminal vertices, and every unused spectator when \(k>2s\). \subsection{Scope} \label{common:subsec-theorem-boundary} The two theorems prove source geometry in Walker's extended cobordism category. They do not prove an overlapping-slide relation, an arbitrary handleslide groupoid, relations among pentagons, stabilization, Cerf completeness, Heegaard invariance, a TQFT construction, WRT invariance, a Habiro lift, or a \(\widehat Z\) comparison. The overlapping three-handle pentagon remains an independent companion theorem. A normalized ordinary \(\mathrm{SO}(3)\)-TQFT scalar for the fixed-site double is recorded only after all source-topological arguments, in \cref{common:sec-ordinary-scalar}. \paragraph{Organization.} \Cref{common:sec-walker-data} fixes the common Walker conventions and separates the two Boolean state spaces. \Cref{fixed:sec-local-calculus,fixed:sec-order-traces,fixed:sec-match-dagger} prove the fixed-site theorem. \Cref{slide:sec-setup,slide:sec:vertices-homology-phasezero,slide:sec:phase-walker-covariance,slide:sec:squares-order-closure} construct and prove the slide cube. The optional ordinary scalar appears in \cref{common:sec-ordinary-scalar}; boundary cases, reusable interfaces, and exact nonclaims are collected in \cref{common:sec-scope-boundary}.