\section{Fixed-site interchange, attachment orders, and completed traces} \label{fixed:sec-order-traces} The preceding local table answers one site at a time; this section explains why those local answers can be assembled in any order. The key issue is not the commutativity of disjoint supports alone, but the preservation of subset tags, endpoint markings, and Walker weights while the history is reordered. First establish the local interchange identities for bare, phased, forward, and reverse arrows while retaining every site tag and weight. Next use these adjacent swaps to compare tagged order words inside their fixed Hom sets; the result is an equality-only closure, not a new global groupoid assertion. Then compute the homological relations and Walker seams for the ordered histories, including both internal and inter-factor phase corrections. Finally replace those ordered histories by the independently constructed simultaneous trace, which is the geometric input used by the later match-count argument. Throughout this section composition is right-factor first. We use the localized arrows proved in the preceding section, with the types \[ J_{i\mid I}^{e}:\Sigma_I^{\mathrm{out}}\longrightarrow \Sigma_{I\cup\{i\}}^{\mathrm{out}}, \qquad R_{i\mid I}^{v}:\Sigma_{I\cup\{i\}}^{\mathrm{out}}\longrightarrow \Sigma_I^{\mathrm{out}}. \] Their actual Walker weights are \[ a_i=c_i+e u_i^+, \qquad b_i'=r_i+v u_i^-, \] where an inactive slope coordinate is absent, not a zero-weight additional arrow. We also use only the already proved elementary handle-arrow dagger identities \[ \bigl(J_{i\mid I}^{e}(c_i,u_i^+)\bigr)^\dagger =R_{i\mid I}^{e}(-c_i,-u_i^+), \qquad \bigl(R_{i\mid I}^{v}(r_i,u_i^-)\bigr)^\dagger =J_{i\mid I}^{v}(-r_i,-u_i^-), \] and the already proved weighted same-site table. In particular, no global dagger formula and no order-independent trace is an input below. For a bit \(q\in\{0,1\}\), put \begin{equation} L_s^q:= \begin{cases} \langle m_s\rangle,&q=0,\\ \langle\ell_s\rangle,&q=1. \end{cases} \label{fixed:eq-phase-line} \end{equation} The sign with which the half-turn sends \(m_s\) to \(\ell_s\) is irrelevant for this one-di\-men\-sional subspace. This notation does not identify the marked mapping classes \(\sigma_s\) and \(\sigma_s^{-1}\). \begin{proposition}[Disjoint-support interchange] \label{fixed:prop-disjoint-support-interchange} Let \(I\subseteq[k]\) and let \(i,j\notin I\) be distinct. For arbitrary independent integers \(r_i,r_j,u_i,u_j\), the following are literal equalities of marked Walker extended cobordisms. The bare reverse handles satisfy \begin{align} &\mathcal R_{i\mid I}^{\mathrm{rev}}(r_i) \mathcal R_{j\mid I\cup\{i\}}^{\mathrm{rev}}(r_j) \notag\\ &\qquad= \mathcal R_{j\mid I}^{\mathrm{rev}}(r_j) \mathcal R_{i\mid I\cup\{j\}}^{\mathrm{rev}}(r_i) :\Sigma_{I\cup\{i,j\}}^{\mathrm{out}} \longrightarrow\Sigma_I^{\mathrm{out}}, \label{fixed:eq-disjoint-bare-reverse} \end{align} with common weight \(r_i+r_j\). On the common high surface, the slope cylinders satisfy \begin{align} &\Theta_{i\mid I\cup\{j\}}(u_i) \Theta_{j\mid I\cup\{i\}}(u_j) \notag\\ &\qquad= \Theta_{j\mid I\cup\{i\}}(u_j) \Theta_{i\mid I\cup\{j\}}(u_i) \in\operatorname{End} (\Sigma_{I\cup\{i,j\}}^{\mathrm{out}}), \label{fixed:eq-disjoint-slopes} \end{align} with common weight \(u_i+u_j\). A slope cylinder also passes through the reverse compression at the other site: \begin{align} &\Theta_{i\mid I}(u_i) \mathcal R_{j\mid I\cup\{i\}}^{\mathrm{rev}}(r_j) \notag\\ &\qquad= \mathcal R_{j\mid I\cup\{i\}}^{\mathrm{rev}}(r_j) \Theta_{i\mid I\cup\{j\}}(u_i) :\Sigma_{I\cup\{i,j\}}^{\mathrm{out}} \longrightarrow\Sigma_{I\cup\{i\}}^{\mathrm{out}}, \label{fixed:eq-disjoint-slope-reverse} \end{align} with common weight \(u_i+r_j\). For a generic attachment state \(I'\subseteq[k]\setminus\{s\}\), define the slope-corrected reverse at a site by \[ \mathcal R_{s\mid I'}^{\sigma}(r_s,u_s) :=\mathcal R_{s\mid I'}^{\mathrm{rev}}(r_s)\Theta_{s\mid I'}(u_s). \] Then \begin{align} &\mathcal R_{i\mid I}^{\sigma}(r_i,u_i) \mathcal R_{j\mid I\cup\{i\}}^{\sigma}(r_j,u_j) \notag\\ &\qquad= \mathcal R_{j\mid I}^{\sigma}(r_j,u_j) \mathcal R_{i\mid I\cup\{j\}}^{\sigma}(r_i,u_i) :\Sigma_{I\cup\{i,j\}}^{\mathrm{out}} \longrightarrow\Sigma_I^{\mathrm{out}}, \label{fixed:eq-disjoint-corrected-reverse} \end{align} with common weight \(r_i+r_j+u_i+u_j\). Every equality fixes the old surface, every attached spectator, and every unused disk, and it preserves the label attached to each weight. \end{proposition} \paragraph{Disjoint supports and the Walker seam.} Because the two active neighborhoods are disjoint, the geometric maps commute on the local pieces. The remaining issue is the Walker seam: old, attached, and unused spectators contribute repeated Lagrangian lines, so only the two active planes require a direct check. \begin{proof} All active maps are supported in disjoint neighborhoods of the \(i\)- and \(j\)-sites. On an already attached spectator plane, the incoming, declared, transported, and outgoing lines are all \(\langle m_s\rangle\), so direct-sum additivity contributes only \[ \mu(\langle m_s\rangle,\langle m_s\rangle, \langle m_s\rangle)=0. \] An unused disk contributes no symplectic plane, and each active map extends by the identity near its boundary circle. It is therefore enough to check the two active sites, while retaining the fixed endpoint parameterizations. The two bare reverse traces remove handles in disjoint labelled regions. Exchanging their critical heights gives a diffeomorphism relative to both boundary parameterizations and preserves each handle label. In the two orders the Walker interface triples are, respectively, \[ \mu(\lambda_{I\cup\{i\}},\lambda_{I\cup\{i\}}, \lambda_{I\cup\{i\}})=0, \qquad \mu(\lambda_{I\cup\{j\}},\lambda_{I\cup\{j\}}, \lambda_{I\cup\{j\}})=0. \] Thus the underlying marked equality lifts with weight \(r_i+r_j\), proving \eqref{fixed:eq-disjoint-bare-reverse}. The two slope mapping classes have disjoint supports and commute. For the order on the left of \eqref{fixed:eq-disjoint-slopes}, the Walker correction splits as \begin{align*} &\mu(\lambda_g,\lambda_g,\lambda_g) +\sum_{s\in I} \mu(\langle m_s\rangle,\langle m_s\rangle, \langle m_s\rangle)\\ &\qquad +\mu(\langle m_i\rangle,\langle m_i\rangle, \langle\ell_i\rangle) +\mu(\langle\ell_j\rangle,\langle m_j\rangle, \langle m_j\rangle)=0. \end{align*} For the opposite order, the last two terms are \[ \mu(\langle\ell_i\rangle,\langle m_i\rangle, \langle m_i\rangle) +\mu(\langle m_j\rangle,\langle m_j\rangle, \langle\ell_j\rangle)=0. \] Hence both products retain weight \(u_i+u_j\), proving \eqref{fixed:eq-disjoint-slopes}. Because \(i\ne j\), the mapping class supported at \(i\) extends by the identity across the reverse compression that removes the \(j\)-handle. This gives the underlying relative-boundary equality in \eqref{fixed:eq-disjoint-slope-reverse}. Its two Walker corrections are \begin{align*} &\mu\bigl((\sigma_{i\mid I\cup\{j\}})_* \lambda_{I\cup\{i,j\}}, \lambda_{I\cup\{i,j\}}, \lambda_{I\cup\{i,j\}}\bigr)=0,\\ &\mu\bigl(\lambda_{I\cup\{i\}}, \lambda_{I\cup\{i\}}, (\sigma_{i\mid I})_*^{-1} \lambda_{I\cup\{i\}}\bigr)=0. \end{align*} Each has a repeated pair, so both sides have weight \(u_i+r_j\). Finally, expand both corrected reverses. The preceding three identities give the typed calculation \begin{align*} &\mathcal R_{i\mid I}^{\mathrm{rev}}(r_i)\Theta_{i\mid I}(u_i) \mathcal R_{j\mid I\cup\{i\}}^{\mathrm{rev}}(r_j) \Theta_{j\mid I\cup\{i\}}(u_j)\\ &\quad= \mathcal R_{i\mid I}^{\mathrm{rev}}(r_i) \mathcal R_{j\mid I\cup\{i\}}^{\mathrm{rev}}(r_j) \Theta_{i\mid I\cup\{j\}}(u_i) \Theta_{j\mid I\cup\{i\}}(u_j)\\ &\quad= \mathcal R_{j\mid I}^{\mathrm{rev}}(r_j) \mathcal R_{i\mid I\cup\{j\}}^{\mathrm{rev}}(r_i) \Theta_{j\mid I\cup\{i\}}(u_j) \Theta_{i\mid I\cup\{j\}}(u_i)\\ &\quad= \mathcal R_{j\mid I}^{\mathrm{rev}}(r_j)\Theta_{j\mid I}(u_j) \mathcal R_{i\mid I\cup\{j\}}^{\mathrm{rev}}(r_i) \Theta_{i\mid I\cup\{j\}}(u_i). \end{align*} The first and last steps use \eqref{fixed:eq-disjoint-slope-reverse}; the middle step uses \eqref{fixed:eq-disjoint-bare-reverse} and \eqref{fixed:eq-disjoint-slopes}. Every interface correction has already been computed, so the common weight is \(r_i+r_j+u_i+u_j\). This proves \eqref{fixed:eq-disjoint-corrected-reverse}. \end{proof} \Needspace{4\baselineskip} \subsection{Heterogeneous adjacent swaps} \begin{theorem}[Fixed-site heterogeneous phase interchange] \label{fixed:thm-interchange} \textup{(R)} Let \(I\subseteq[k]\) and let \(i,j\notin I\) be distinct. For every \((\eta_i,\eta_j)\in\{0,1\}^2\), one has the typed equality \begin{align} &R_{i\mid I}^{\eta_i} R_{j\mid I\cup\{i\}}^{\eta_j} \notag\\ &\qquad= R_{j\mid I}^{\eta_j} R_{i\mid I\cup\{j\}}^{\eta_i} :\Sigma_{I\cup\{i,j\}}^{\mathrm{out}} \longrightarrow\Sigma_I^{\mathrm{out}}. \label{fixed:eq-I0} \end{align} The common Walker weight is \begin{equation} b_i'+b_j'=r_i+r_j+\eta_i u_i^-+\eta_j u_j^-. \label{fixed:eq-I0-weight} \end{equation} Here and below a phase-zero factor has only its handle-weight argument, whereas a phase-one factor has its handle and active slope-weight arguments. \par\smallskip \textup{(F)} Under the same hypotheses, for every \((\epsilon_i,\epsilon_j)\in\{0,1\}^2\), \begin{align} &J_{j\mid I\cup\{i\}}^{\epsilon_j} J_{i\mid I}^{\epsilon_i} \notag\\ &\qquad= J_{i\mid I\cup\{j\}}^{\epsilon_i} J_{j\mid I}^{\epsilon_j} :\Sigma_I^{\mathrm{out}} \longrightarrow\Sigma_{I\cup\{i,j\}}^{\mathrm{out}}. \label{fixed:eq-I1} \end{align} The common Walker weight is \begin{equation} a_i+a_j=c_i+c_j+\epsilon_i u_i^++\epsilon_j u_j^+. \label{fixed:eq-I1-weight} \end{equation} The phase-dependent argument convention is the one stated after \eqref{fixed:eq-I0}; in particular no inactive \(u_s^+\) is passed to a phase-zero arrow. \end{theorem} \paragraph{Four local phase cases and source reversal.} Prove the reverse interchange in the four phase cases, keeping the subset tags visible. The mixed cases move a slope cylinder only across the different-site reverse handle; source reversal then supplies the forward interchange with the correct factor order and weight. \begin{proof}[Proof of the reverse relations] We prove the four cases before using any order theorem. If \((\eta_i,\eta_j)=(0,0)\), equation \eqref{fixed:eq-I0} is \eqref{fixed:eq-disjoint-bare-reverse}, with its full subset tags and common weight \(r_i+r_j\). If \((\eta_i,\eta_j)=(1,1)\), expand both phase-one arrows as \(R^1=\mathcal R^{\mathrm{rev}} \Theta\). Equation \eqref{fixed:eq-disjoint-corrected-reverse}, with \(u_s=u_s^-\), gives the required equality and common weight \(r_i+r_j+u_i^-+u_j^-\). Thus this case uses the exact slope words and not an assertion that a half-turn squares to the identity. For \((\eta_i,\eta_j)=(0,1)\), all subset tags are visible in the following calculation: \begin{align*} &\mathcal R_{i\mid I}^{\mathrm{rev}}(r_i) \mathcal R_{j\mid I\cup\{i\}}^{\mathrm{rev}}(r_j) \Theta_{j\mid I\cup\{i\}}(u_j^-) \\ &\quad= \mathcal R_{j\mid I}^{\mathrm{rev}}(r_j) \mathcal R_{i\mid I\cup\{j\}}^{\mathrm{rev}}(r_i) \Theta_{j\mid I\cup\{i\}}(u_j^-) &&\text{by \eqref{fixed:eq-disjoint-bare-reverse}} \\ &\quad= \mathcal R_{j\mid I}^{\mathrm{rev}}(r_j) \Theta_{j\mid I}(u_j^-) \mathcal R_{i\mid I\cup\{j\}}^{\mathrm{rev}}(r_i) &&\text{by \eqref{fixed:eq-disjoint-slope-reverse}, read in reverse} \\ &\quad= R_{j\mid I}^{1}(r_j,u_j^-) R_{i\mid I\cup\{j\}}^{0}(r_i). \end{align*} The second equality slides the slope supported at \(j\) only past the different-site reverse supported at \(i\). Every line has type \(\Sigma_{I\cup\{i,j\}}^{\mathrm{out}}\to \Sigma_I^{\mathrm{out}}\) and exact weight \(r_i+r_j+u_j^-\). For \((\eta_i,\eta_j)=(1,0)\), the other mixed calculation is \begin{align*} &\mathcal R_{i\mid I}^{\mathrm{rev}}(r_i) \Theta_{i\mid I}(u_i^-) \mathcal R_{j\mid I\cup\{i\}}^{\mathrm{rev}}(r_j) \\ &\quad= \mathcal R_{i\mid I}^{\mathrm{rev}}(r_i) \mathcal R_{j\mid I\cup\{i\}}^{\mathrm{rev}}(r_j) \Theta_{i\mid I\cup\{j\}}(u_i^-) &&\text{by \eqref{fixed:eq-disjoint-slope-reverse}} \\ &\quad= \mathcal R_{j\mid I}^{\mathrm{rev}}(r_j) \mathcal R_{i\mid I\cup\{j\}}^{\mathrm{rev}}(r_i) \Theta_{i\mid I\cup\{j\}}(u_i^-) &&\text{by \eqref{fixed:eq-disjoint-bare-reverse}} \\ &\quad= R_{j\mid I}^{0}(r_j) R_{i\mid I\cup\{j\}}^{1}(r_i,u_i^-). \end{align*} Here the first equality slides the \(i\)-slope only past the different-site \(j\)-reverse. Every line again has the required endpoints, now with exact weight \(r_i+r_j+u_i^-\). These four calculations prove both \eqref{fixed:eq-I0} and \eqref{fixed:eq-I0-weight} without a same-site cancellation, a left-inverse relation, or an order-independence premise. Indeed, before any surrounding history is added, the outer reverse seam on the two sides is respectively \[ \mu\bigl(\lambda_{I\cup\{i\}},\lambda_{I\cup\{i\}}, \lambda_I\oplus L_i^{\eta_i}\bigr)=0, \qquad \mu\bigl(\lambda_{I\cup\{j\}},\lambda_{I\cup\{j\}}, \lambda_I\oplus L_j^{\eta_j}\bigr)=0. \] The phase-one internal seams were proved in the localized phase calculation. Thus the displayed sums are already the actual weights; the proposition below computes the additional mixed spectator lines created by a surrounding history. \end{proof} \begin{proof}[Proof of the forward relations] The forward relation is not proved by introducing a new local calculation. Apply source reversal to the already established reverse relation, negate the constituent weights, and use the elementary dagger identity at each tagged site. This preserves the distinction between a slope reversal and a coordinate rewrite. Apply the already proved reverse relation \eqref{fixed:eq-I0} with \(\eta_s=\epsilon_s\), \(r_s=-c_s\), and, when the phase is active, \(u_s^-=-u_s^+\), for \(s=i,j\). This is legitimate because the reverse relations have already been proved in all four cases. Actual source orientation reversal reverses the product order and negates its Walker weight. The already proved elementary identity for either \(s\in\{i,j\}\) and its relevant subset tag \(I'\) is \[ \bigl(R_{s\mid I'}^{\epsilon_s}\bigr)^\dagger =J_{s\mid I'}^{\epsilon_s} \] Assign the handle weight \(-c_s\) and the active slope weight \(-u_s^+\); after dagger these weights are \(c_s,u_s^+\). With these weights, the elementary identity turns the dagger of the left side of \eqref{fixed:eq-I0} into the left side of \eqref{fixed:eq-I1}, and similarly on the right. The common reverse weight \(-a_i-a_j\) becomes \(a_i+a_j\). No generator-level replacement of \(\Theta(u)^\dagger\) by \(\Theta(-u)\), and no global dagger theorem, occurs in this argument. \end{proof} \begin{proposition}[Mixed spectator Maslov terms] \label{fixed:prop-spectator-maslov} Every Walker interface used in the reverse and forward parts of Theorem~\ref{fixed:thm-interchange}, including an occurrence inside a longer phase history, has total Maslov correction zero. This remains true with the entire old surface, every attached site in \(I\), and every unused disk retained. \end{proposition} \paragraph{Spectator splitting and Maslov cancellation.} Decompose every intermediate interface into old, active, spectator, and unused summands. Repeated pairs vanish in the old and spectator summands; the active triples are exactly the local phase triples already calculated. This reduces every surrounding-history correction to the local zero calculation. \begin{proof} All maps in an active swap are supported in the \(i\)- and \(j\)-regions. By the chosen splitting of first homology, an interface decomposes into the old-surface summand, the active \(i,j\) planes that are present at that interface, and one plane for every attached spectator. The old-surface term is \[ \mu(\lambda_g,\lambda_g,\lambda_g)=0. \] An unused disk has not acquired a torus plane and contributes no symplectic summand. We now calculate rather than infer the attached-spectator terms. A phase history supported at a spectator \(s\in I\) can put either \(L_s^0\) or \(L_s^1\) on each transported side, while the declared middle line is always \(\langle m_s\rangle\): a phase-zero local word fixes the meridian line, a phase-one local word sends it to the longitude line, and every different-site factor is the identity on this plane. Thus the complete list is \begin{equation} \begin{array}{c|c} (q,q')&\text{spectator triple}\\ \hline (0,0)&(\langle m_s\rangle,\langle m_s\rangle, \langle m_s\rangle)\\ (0,1)&(\langle m_s\rangle,\langle m_s\rangle, \langle\ell_s\rangle)\\ (1,0)&(\langle\ell_s\rangle,\langle m_s\rangle, \langle m_s\rangle)\\ (1,1)&(\langle\ell_s\rangle,\langle m_s\rangle, \langle\ell_s\rangle). \end{array} \label{fixed:eq-I2-spectator-table} \end{equation} Every row has a repeated pair, so every spectator term vanishes. In particular, this explicitly includes \[ \mu(\langle m_s\rangle,\langle m_s\rangle,\langle\ell_s\rangle),\qquad \mu(\langle\ell_s\rangle,\langle m_s\rangle,\langle m_s\rangle),\qquad \mu(\langle\ell_s\rangle,\langle m_s\rangle,\langle\ell_s\rangle). \] For completeness, the active-plane seams in the displayed reverse words are also explicit. A bare reverse gives \((m,m,m)\). At a slope--different-site-reverse slide the two sides give \((\ell,m,m)\) and \((m,m,\ell)\), respectively. Expanding the corrected interchange introduces only these triples and the two-slope possibility \((\ell,m,\ell)\). The forward seams obtained by orientation reversal have the same list with first and third entries exchanged. Hence each active term also has a repeated pair. In particular, the mixed \(01\) and \(10\) proofs above use only the zero-correction interfaces \eqref{fixed:eq-disjoint-slope-reverse} and \eqref{fixed:eq-disjoint-bare-reverse}, while the \(11\) proof uses the zero-correction expansion of \eqref{fixed:eq-disjoint-corrected-reverse}. Direct-sum additivity now gives, at every interface, \begin{equation} \mu_{\mathrm{total}} =\mu_{\mathrm{old}}+\mu_i+\mu_j +\sum_{s\in I}\mu_s=0, \label{fixed:eq-I2} \end{equation} with a missing active plane simply omitted. Thus the weights used in \eqref{fixed:eq-I0-weight} and \eqref{fixed:eq-I1-weight} are the actual Walker weights on the complete chart, not merely the empty-spectator weights. \end{proof} \subsection{Order words and equality-only Coxeter closure} To pass from one adjacent interchange to arbitrary attachment histories, we need order words that keep each phase bit and constituent weight attached to its disk label. The following notation also makes the fixed source and target of the two directions explicit. \begin{definition}[Tagged phased order words] \label{fixed:def-order-words} For \(\alpha=(\alpha_1,\ldots,\alpha_k)\in S_k\), set \begin{equation} I_a^\alpha:=\{\alpha_1,\ldots,\alpha_a\}, \qquad I_0^\alpha:=\varnothing. \label{fixed:eq-O0} \end{equation} Define \begin{align} J_\alpha^\epsilon(\mathbf c,\mathbf u^+) &:={J}_{\alpha_k\mid I_{k-1}^\alpha}^{\epsilon_{\alpha_k}} \cdots {J}_{\alpha_1\mid\varnothing}^{\epsilon_{\alpha_1}}: \Sigma_\varnothing^{\mathrm{out}}\longrightarrow \Sigma_{[k]}^{\mathrm{out}}, \label{fixed:eq-O1}\\ R_\alpha^\eta(\mathbf r,\mathbf u^-) &:={R}_{\alpha_1\mid\varnothing}^{\eta_{\alpha_1}} \cdots {R}_{\alpha_k\mid I_{k-1}^\alpha}^{\eta_{\alpha_k}}: \Sigma_{[k]}^{\mathrm{out}}\longrightarrow \Sigma_\varnothing^{\mathrm{out}}. \label{fixed:eq-O2} \end{align} Every phase bit and every constituent weight in these expressions remains attached to its disk label. The tag \(\alpha\) on the reverse word is its matching attachment word; chronologically, the right-factor-first reverse word removes \(\alpha_k,\ldots,\alpha_1\). \end{definition} \begin{corollary}[Phased order independence] \label{fixed:cor-order-independence} For each fixed phase vector and all label-attached weights, the expressions in Definition \ref{fixed:def-order-words} have literal common values \begin{align} J_\alpha^\epsilon(\mathbf c,\mathbf u^+) &=:J_{[k]}^\epsilon(\mathbf c,\mathbf u^+), \label{fixed:eq-O3}\\ R_\alpha^\eta(\mathbf r,\mathbf u^-) &=:R_{[k]}^\eta(\mathbf r,\mathbf u^-), \label{fixed:eq-O4} \end{align} in their two fixed-endpoint Hom sets, for every \(\alpha\in S_k\). \end{corollary} \begin{proof} Any two order words are joined by finitely many adjacent transpositions. Suppose positions \(a,a+1\) contain \(i,j\), and let \(I\) be the set of the labels in the preceding positions. The affected block of the forward word is exactly \[ J_{j\mid I\cup\{i\}}^{\epsilon_j} J_{i\mid I}^{\epsilon_i}, \] so Theorem \ref{fixed:thm-interchange} replaces it by the block with \(i,j\) interchanged. The affected algebraic block of the reverse word is exactly \[ R_{i\mid I}^{\eta_i} R_{j\mid I\cup\{i\}}^{\eta_j}, \] so Theorem \ref{fixed:thm-interchange} performs the matching replacement. All outside factors have literally matching tagged endpoints. Proposition \ref{fixed:prop-spectator-maslov} supplies the full Walker weight calculation when the replacement sits inside the longer history. A swap therefore moves labels, phases, and their constituent weights together and does not change the sum. Finite induction proves \eqref{fixed:eq-O3}--\eqref{fixed:eq-O4}. \end{proof} \begin{proposition}[Equality-only Coxeter closure] \label{fixed:prop-coxeter-closure} For the forward words and reverse words separately, and for each fixed phase vector, let \(\tau_a\) denote only the equality replacement interchanging the entries in positions \(a,a+1\). These replacements close around \begin{equation} \tau_a^2=1, \qquad \tau_a\tau_b=\tau_b\tau_a\quad(|a-b|>1), \qquad \tau_a\tau_{a+1}\tau_a=\tau_{a+1}\tau_a\tau_{a+1}. \label{fixed:eq-O5} \end{equation} These are loops of literal equalities of expressions. They retain no swap cylinder, comparison isotopy, or \(S_k\)-action. \end{proposition} \begin{proof} Check the three Coxeter-shaped relations at the level of labelled expressions. Involutivity reverses one equality, distant swaps involve disjoint positions, and the braid uses the same three labels with different subset tags. Since these are equality paths, no swap cobordism or permutation action is created. For \(\tau_a^2\), the second step traverses the same instance of \eqref{fixed:eq-I0} or \eqref{fixed:eq-I1} backward. The preceding set, both phases, and all label-attached weights are unchanged. If \(|a-b|>1\), the two swaps involve disjoint pairs of positions. Either order produces the same four labeled factors. Every subset tag is the set of labels preceding a position, not an ordering of that set; consequently the two routes have identical tagged endpoints and weights. For the braid, let the three consecutive labels be \(i,j,t\), and let \(I\) be the set of earlier labels. The first route has label orders \[ (i,j,t)\longrightarrow(j,i,t)\longrightarrow(j,t,i) \longrightarrow(t,j,i), \] whereas the second has \[ (i,j,t)\longrightarrow(i,t,j)\longrightarrow(t,i,j) \longrightarrow(t,j,i). \] The active pair and exact base tag on the two routes are \[ \begin{array}{c|ccc} \text{first route}&(i,j\mid I)& (i,t\mid I\cup\{j\})&(j,t\mid I)\\ \text{second route}&(j,t\mid I\cup\{i\})& (i,t\mid I)&(i,j\mid I\cup\{t\}). \end{array} \] Thus every edge is a correctly typed instance of \eqref{fixed:eq-I0} or \eqref{fixed:eq-I1}; phases and weights follow \(i,j,t\), not the three positions. Both routes end at the same expression, so transitivity of literal equality closes the diagram. For \(k=1\) there is no generator; for \(k=2\) only the involutive loop is present. \end{proof} \subsection{Ordered histories before a common trace} For a cobordism \(W:\Sigma_-\to\Sigma_+\), write \begin{equation} \mathscr N(W):=\ker\!\left( H_1(\overline{\Sigma_-}\sqcup\Sigma_+;\mathbb Q) \longrightarrow H_1(W;\mathbb Q)\right) \label{fixed:eq-homological-relation} \end{equation} for its homological Lagrangian relation, and use \(\mathscr N(W)_*\) and \(\mathscr N(W)^*\) for relational image and inverse image. For a phase assignment \(q\) on \(I\), put \begin{equation} \lambda_I^q:=\lambda_g\oplus \bigoplus_{s\in I}L_s^{q_s}. \label{fixed:eq-phased-lagrangian} \end{equation} \begin{lemma}[One-edge transport and internal phase seams] \label{fixed:lem-one-edge-history} For \(i\notin I\) and \(e,v\in\{0,1\}\), the underlying three-manifold of each phase arrow is the corresponding compression body. There is an identification \begin{equation} H_1(J_{i\mid I}^{e};\mathbb Q) \cong H_1(\Sigma_I^{\mathrm{out}};\mathbb Q)\oplus\mathbb Qx_i, \label{fixed:eq-one-edge-homology} \end{equation} and the same statement holds for \(R_{i\mid I}^{v}\). Put \[ P_i:=\operatorname{Span}(m_i,\ell_i), \] and let \(j_0:P_i\to\mathbb Qx_i\) be the active-plane part of the bare high-boundary inclusion. Under the homology identifications above, the forward-target and reverse-source inclusions are distinguished by \begin{equation} j_{J,0}=j_0,\qquad j_{J,1}=j_0\circ(\sigma_{i\mid I})_*^{-1}, \qquad j_{R,0}=j_0,\qquad j_{R,1}=j_0\circ(\sigma_{i\mid I})_*. \label{fixed:eq-phase-inclusion-relations} \end{equation} Explicitly, \begin{equation} \begin{array}{c|cc} &m_i&\ell_i\\ \hline j_{J,0}&0&x_i\\ j_{J,1}&x_i&0\\ j_{R,0}&0&x_i\\ j_{R,1}&-x_i&0 \end{array}. \label{fixed:eq-boundary-map} \end{equation} Invariantly, \begin{equation} \begin{gathered} \ker(j_{J,e})=L_i^e,\qquad \ker(j_{R,v})=L_i^v,\\ P_i/L_i^e\xrightarrow[\ j_{J,e}\ ]{\ \cong\ }\mathbb Qx_i,\\ P_i/L_i^v\xrightarrow[\ j_{R,v}\ ]{\ \cong\ }\mathbb Qx_i. \end{gathered} \label{fixed:eq-phased-kernel-quotients} \end{equation} The exact one-edge transports are \begin{align} \mathscr N(J_{i\mid I}^{e})_*\lambda_I &=\lambda_I\oplus L_i^e, & \mathscr N(J_{i\mid I}^{e})^*\lambda_{I\cup\{i\}} &=\lambda_I, \label{fixed:eq-forward-edge-transport}\\ \mathscr N(R_{i\mid I}^{v})_*\lambda_{I\cup\{i\}} &=\lambda_I, & \mathscr N(R_{i\mid I}^{v})^*\lambda_I &=\lambda_I\oplus L_i^v. \label{fixed:eq-reverse-edge-transport} \end{align} For a phase-one forward or reverse arrow, its sole internal slope/handle seam has Maslov index zero; hence its actual weights are \(c_i+u_i^+\) and \(r_i+u_i^-\), respectively. \end{lemma} \paragraph{Core classes, phase lines, and seam cancellation.} The one-edge calculation separates the new handle plane from the old and spectator summands. The bare compression relation supplies the core class, the phase cylinder precomposes the distinguished boundary inclusion by the slope action, and the resulting kernel and quotient are the displayed phased lines. Composing that relation with the compression relation gives the transport formulas, while the internal Walker seam is reduced separately to a repeated Lagrangian pair. \begin{proof} A slope cylinder changes neither the underlying three-manifold nor its homology. For the bare compression body, the handle core contributes \(x_i\), the co-core bounds \(m_i\), and the framed longitude maps to \(x_i\). Thus \(j_0(m_i)=0\) and \(j_0(\ell_i)=x_i\). The chosen exact slope action is \[ (\sigma_{i\mid I})_*m_i=-\ell_i,\qquad (\sigma_{i\mid I})_*\ell_i=m_i, \] and hence \[ (\sigma_{i\mid I})_*^{-1}m_i=\ell_i,\qquad (\sigma_{i\mid I})_*^{-1}\ell_i=-m_i. \] The forward phase cylinder changes the target inclusion by precomposition with \((\sigma_{i\mid I})_*^{-1}\), whereas the reverse phase cylinder changes the source inclusion by precomposition with \((\sigma_{i\mid I})_*\). This proves \eqref{fixed:eq-one-edge-homology}, \eqref{fixed:eq-phase-inclusion-relations}, and the complete table \eqref{fixed:eq-boundary-map}. Its kernels are exactly the phased lines \(L_i^e\) and \(L_i^v\), and each remaining one-dimensional quotient maps isomorphically to \(\mathbb Qx_i\), proving \eqref{fixed:eq-phased-kernel-quotients}. Composing these distinguished inclusions, equivalently the graph relation of the appropriate slope cylinder, with the compression relation gives \eqref{fixed:eq-forward-edge-transport} and \eqref{fixed:eq-reverse-edge-transport}; every old and spectator summand is fixed. Notice that this is a statement about Lagrangian lines, not an equality \(\sigma_i=\sigma_i^{-1}\). For \(J^1=\Theta\mathcal C\), the internal seam is \[ \mu\bigl(\lambda_{I\cup\{i\}}, \lambda_{I\cup\{i\}}, \lambda_I\oplus\langle\ell_i\rangle\bigr)=0. \] For \(R^1=\mathcal R^{\mathrm{rev}} \Theta\), it is \[ \mu\bigl(\lambda_I\oplus\langle\ell_i\rangle, \lambda_{I\cup\{i\}}, \lambda_{I\cup\{i\}}\bigr)=0. \] The entries agree on the old and spectator summands, and on the active plane the displayed triples are respectively \((m,m,\ell)\) and \((\ell,m,m)\). Repeated-pair vanishing and direct-sum additivity prove the claim and recover the active-weight formulas rather than assuming them. \end{proof} \begin{proposition}[Every mixed ordered history] \label{fixed:prop-ordered-histories} For every phase vector and every order \(\alpha\), calculated separately before introducing a simultaneous trace, define the partial words \begin{align} J_{\alpha,\leq a}^\epsilon &:= J_{\alpha_a\mid I_{a-1}^\alpha}^{\epsilon_{\alpha_a}} \cdots J_{\alpha_1\mid\varnothing}^{\epsilon_{\alpha_1}}, \label{fixed:eq-forward-partial-word}\\ R_{\alpha,\geq a}^\eta &:= R_{\alpha_a\mid I_{a-1}^\alpha}^{\eta_{\alpha_a}} \cdots R_{\alpha_k\mid I_{k-1}^\alpha}^{\eta_{\alpha_k}}. \label{fixed:eq-reverse-partial-word} \end{align} At all intermediate interfaces their exact transports are \begin{align} \mathscr N(J_{\alpha,\leq a}^\epsilon)_*\lambda_g &=\lambda_g\oplus \bigoplus_{t=1}^aL_{\alpha_t}^{\epsilon_{\alpha_t}}, & \mathscr N(J_{\alpha,\leq a}^\epsilon)^*\lambda_{I_a^\alpha} &=\lambda_g, \label{fixed:eq-forward-partial-transport}\\ \mathscr N(R_{\alpha,\geq a}^\eta)_*\lambda_{[k]} &=\lambda_{I_{a-1}^\alpha}, & \mathscr N(R_{\alpha,\geq a}^\eta)^*\lambda_{I_{a-1}^\alpha} &=\lambda_{I_{a-1}^\alpha}\oplus \bigoplus_{t=a}^kL_{\alpha_t}^{\eta_{\alpha_t}}. \label{fixed:eq-reverse-partial-transport} \end{align} Moreover, \begin{align} H_1(J_\alpha^\epsilon;\mathbb Q) &\cong H_1(\Sigma_g^{\mathrm{out}};\mathbb Q) \oplus\bigoplus_{a=1}^k\mathbb Qx_{\alpha_a}, &\beta_1(J_\alpha^\epsilon)&=2g+k, \label{fixed:eq-forward-ordered-homology}\\ H_1(R_\alpha^\eta;\mathbb Q) &\cong H_1(\Sigma_g^{\mathrm{out}};\mathbb Q) \oplus\bigoplus_{a=1}^k\mathbb Qx_{\alpha_a}, &\beta_1(R_\alpha^\eta)&=2g+k. \label{fixed:eq-reverse-ordered-homology} \end{align} Writing \(\operatorname{wt}\) for the integer Walker weight, the total weights are \begin{equation} \operatorname{wt}(J_\alpha^\epsilon)=\sum_{i=1}^k a_i, \qquad \operatorname{wt}(R_\alpha^\eta)=\sum_{i=1}^k b_i'. \label{fixed:eq-T0} \end{equation} All intermediate and endpoint markings are the fixed label-preserving markings. \end{proposition} \paragraph{Induction over the labelled sites.} Compute each ordered history before invoking order independence. Induction adds one independent core class per labelled site and transports the phase line through the next interface; repeated-pair Maslov triples make every inter-factor correction vanish. The total weight is therefore the sum of the independently attached local weights. \begin{proof} Iterating \eqref{fixed:eq-one-edge-homology} in the literal order \(\alpha_1,\ldots,\alpha_k\) adds one independent core class at each distinctly labeled site and no relation between disjoint sites. Reversing the compression bodies leaves the same underlying homology calculation. This proves \eqref{fixed:eq-forward-ordered-homology} and \eqref{fixed:eq-reverse-ordered-homology} separately for each \(\alpha\). Induction using \eqref{fixed:eq-forward-edge-transport} and \eqref{fixed:eq-reverse-edge-transport} gives \eqref{fixed:eq-forward-partial-transport} and \eqref{fixed:eq-reverse-partial-transport}; different-site factors fix every previously created phase line. We next compute every inter-factor seam. For \(a=2,\ldots,k\), immediately before the \(a\)-th forward attachment, the relational image of the initial Lagrangian is \[ \lambda_g\oplus \bigoplus_{t