% Provenance: corrected synthesis of stage 24. \section{Raw and Saturated Slide Quotients} \label{p4:retained:ch:raw-saturated} The chamber data become a closed-manifold construction only after quotienting by every successive attaching-slide relation. This section compares the raw integral quotient with its generic saturation and tracks the torsion that can appear at cyclotomic specialization. There are two natural polynomial relation lattices: the relations inherited from the genuine Laurent skein module and their saturation inside the generic fraction-field slide subspace. They agree generically but can differ by cyclotomic torsion, so the distinction must be made before root specialization. First form the complete geometric filling quotient by attaching the two-handles and pulling each later slide kernel back through the preceding quotient. Then compare the polynomial relation lattice with its generic saturation inside the localized slide space. The coefficient-ring obstruction explains why localization cannot be performed before the joint disk--cyclotomic regulator is fixed; finite-stage quotients must instead remain inside the embedded polynomial lattice. Finally, transport the relation-before-reduction construction along verified degrees and state the torsion and specialization limits separately. \subsection{The complete generic filling quotient} Fix a finite genus and a Heegaard diagram $\mathcal D=(\Sigma_g,\boldsymbol\alpha,\boldsymbol\beta)$. Put \[ R=\mathbb Z[A,A^{-1}],\qquad M_R=\operatorname{Sk}_A(H_\alpha;R),\qquad V=\mathbb Q(A)\otimes_RM_R. \] Attach the $\beta$-two-handles successively. Pull every later slide kernel back to $M_R$ through the preceding quotient maps, and let $\mathcal J_{\mathcal D}^R$ be the sum of all these pulled-back kernels. Set \[ \mathcal I_{\mathcal D}^{\FA}=\FA\mathcal J_{\mathcal D}^R\subset V, \qquad \FA=\mathbb Q(A). \] \begin{theorem}[Complete successive-slide quotient] \label{p4:retained:thm:c16-generic-quotient} In every finite genus, \[ M_R/\mathcal J_{\mathcal D}^R \cong\operatorname{Sk}_A(M_\mathcal D;R), \qquad V/\mathcal I_{\mathcal D}^{\FA} \cong\operatorname{Sk}_A(M_\mathcal D;\FA). \] A restricted family of slides need not generate the kernel. This warning is essential even for connected sums of handlebodies \cite{BakshiPrzytycki}. \end{theorem} \begin{proof} The quotient is built in the order in which the two-handles are attached. The one-handle theorem supplies each full slide kernel, pullback records the earlier quotient maps, and flat localization gives the generic statement. Only after this complete geometric quotient is fixed do the raw and saturated polynomial relation lattices become comparable. For one two-handle, the standard handle-attachment theorem \cite{Przytycki} identifies the surjection and its full slide kernel. Apply it in order and pull every later kernel back through the preceding surjections. The final three-handle induces an isomorphism. Localization to $\FA$ is flat and gives the generic vector-space statement. \end{proof} The pointed uncompleted quotient is invariant under Heegaard moves because it is the filled skein module. This alone says nothing about the polynomial chamber height, its completion, a cap, or the anomaly line. \subsection{The coefficient-ring obstruction} Set \[ \Lambda=\mathbb Z[A],\qquad q=A^2, \qquad \mathfrak j_N=A^{2N}(q;q)_N\quad(N\geq0), \qquad \mathfrak j_0=1. \] Then \[ \mathfrak j_{N+1}=A^2(1-q^{N+1})\mathfrak j_N \qquad(N\geq0). \] \begin{proposition}[No naive common skein base change] \label{p4:retained:prop:c16-ring-obstruction} The finite-stage ring $\Lambda/(\mathfrak j_N)$ is not an admissible coefficient ring for a direct application of the ordinary Kauffman-bracket two-handle theorem, because the image of $A$ is not invertible. If $A$ is localized first, then \[ \mathfrak j_N\Lambda[A^{-1}] =(q;q)_N\Lambda[A^{-1}], \] so the disk factor $A^{2N}$ disappears. Thus naive skein base change cannot retain the joint disk--cyclotomic regulator. \end{proposition} \begin{proof} The coefficient-ring obstruction is a noncommuting-operations check. In the polynomial quotient the disk parameter remains a zero divisor, whereas localizing to use the Laurent skein relation makes the disk monomial a unit. The finite stage must therefore be formed inside the already embedded polynomial lattice, keeping disk and cyclotomic control simultaneously. In $\Lambda/(\mathfrak j_N)$ the nonzero class of $A$ is killed by $A^{2N-1}(q;q)_N$, so it is a zero divisor and cannot be a unit. After localization $A^{2N}$ is a unit. The Kauffman relation uses both $A$ and $A^{-1}$, while disk order requires $A$ to remain a nonunit; the operations cannot be interchanged. \end{proof} Accordingly, the finite stages below are abstract quotients of polynomial lattices already embedded in the generic $\FA$-skein space. They are not called Kauffman skein modules over $\Lambda/(\mathfrak j_N)$. \subsection{Diagram-labeled degrees and two relation lattices} Let $\gamma$ be a verified recoupling/bridge degree with generic operator $U_\gamma$ and transported lattice $L_{\gamma,\alpha}=U_\gamma L_{0,\alpha}$ for a fixed source chamber $\alpha$. Carry the filling with the degree: \[ \mathcal J_{\mathcal D,\gamma}^R =U_\gamma\mathcal J_{\mathcal D}^R, \qquad \mathcal I_{\mathcal D,\gamma}^{\FA} =U_\gamma\mathcal I_{\mathcal D}^{\FA}. \] The base label $\mathcal D$ remains fixed; the current geometric diagram is $\gamma\mathcal D$. Replacing it by an independently chosen diagram would change the filling problem. \begin{definition}[Raw and saturated slide lattices] \label{p4:retained:def:c16-two-lattices} Set \begin{align*} \mathscr R_{\mathcal D,\gamma,\alpha} &=L_{\gamma,\alpha}\cap \mathcal J_{\mathcal D,\gamma}^R,\\ \mathscr S_{\mathcal D,\gamma,\alpha} &=L_{\gamma,\alpha}\cap \mathcal I_{\mathcal D,\gamma}^{\FA}, \end{align*} and \[ Q^{\rm raw}_{\mathcal D,\gamma,\alpha} =L_{\gamma,\alpha}/\mathscr R_{\mathcal D,\gamma,\alpha}, \qquad Q^{\rm sat}_{\mathcal D,\gamma,\alpha} =L_{\gamma,\alpha}/\mathscr S_{\mathcal D,\gamma,\alpha}. \] The raw lattice remembers genuine Laurent slide relations. The saturated lattice is the kernel of the map to the generic filled $\FA$-skein space. Both intersections are taken before disk or root reduction. \end{definition} \begin{proposition}[Raw torsion and generic saturation] \label{p4:retained:prop:c16-raw-sat} There are inclusions and an exact sequence \[ \mathscr R\subset\mathscr S\subset L, \qquad 0\longrightarrow\mathscr S/\mathscr R \longrightarrow Q^{\rm raw} \longrightarrow Q^{\rm sat}\longrightarrow0. \] The two relation lattices have the same $\FA$-span. Hence $\mathscr S/\mathscr R$ is polynomial torsion and may carry cyclotomic information. \end{proposition} \begin{proof} Raw and saturated relations are compared by denominator clearing inside the generic slide subspace. A saturated relation has a finite fraction-field expression in raw generators; one common polynomial multiplier returns it to the raw lattice. This exposes the quotient torsion as the finite specialization defect rather than identifying the two relation systems. For $s\in\mathscr S$, express $s$ as a finite $\FA$-linear combination of raw slide elements. Clearing all rational and Laurent denominators gives a nonzero $f\in\Lambda$ with $fs\in\mathcal J^R$. Since $s\in L$, also $fs\in L$, so $fs\in\mathscr R$. This proves equality of $\FA$-spans; the exact sequence is the third isomorphism theorem. \end{proof} \begin{lemma}[Saturation and torsion-freeness] \label{p4:retained:lem:c16-saturated} If $0\neq f\in\Lambda$, $x\in L$, and $fx\in\mathscr S$, then $x\in\mathscr S$. Consequently $Q^{\rm sat}$ is torsion-free over $\Lambda$. \end{lemma} \begin{proof} The polynomial $f$ is invertible in $\FA$, and $\mathcal I^{\FA}$ is a $\FA$-subspace. Thus $fx\in\mathcal I^{\FA}$ implies $x=f^{-1}(fx)\in\mathcal I^{\FA}$. Intersecting with $L$ proves the first claim, which is equivalent to torsion-freeness of the quotient. \end{proof} Torsion-free is the correct conclusion; $\mathbb Z[A]$ is not a PID, so freeness is not inferred. \begin{theorem}[Exact quotient transport] \label{p4:retained:thm:c16-transport} For every verified arrow $a:\gamma\to a\gamma$, the map $U_a$ carries both $\mathscr R$ and $\mathscr S$ exactly to their counterparts and induces $\Lambda$-linear isomorphisms \[ Q^\square_{\mathcal D,\gamma,\alpha} \xrightarrow{\sim} Q^\square_{\mathcal D,a\gamma,\alpha}, \qquad \square\in\{\mathrm{raw},\mathrm{sat}\}. \] Pure recoupling substitutions with the same endpoints induce the same map. \end{theorem} \begin{proof} Transport is exact because the same verified arrow carries the chamber lattice and the pulled-back filling relations. The identity $U_a(L\cap\mathscr X)=U_aL\cap U_a\mathscr X$ transfers both intersections, and the recoupling groupoid then supplies endpoint path independence. For either carried relation space $\mathscr X$, \[ U_a(L\cap\mathscr X)=U_aL\cap U_a\mathscr X. \] This proves exact relation transport and hence the quotient map. Pure recoupling path independence follows from the common underlying skein realization of Section~\ref{p4:retained:ch:recoupling-groupoid}. \end{proof} The directed distributive structure belongs to the chamber index. Images of intersections need not equal intersections of quotient images, so no unproved distributive law for quotient submodules is asserted. \subsection{Relation-before-reduction towers} For $N\geq1$, define \[ Q_{\mathcal D,\gamma,\alpha}^{\rm raw,(N)}= \frac{L_{\gamma,\alpha}} {\mathscr R_{\mathcal D,\gamma,\alpha} +\mathfrak j_NL_{\gamma,\alpha}}, \] \[ Q_{\mathcal D,\gamma,\alpha}^{\rm sat,(N)}= \frac{L_{\gamma,\alpha}} {\mathscr S_{\mathcal D,\gamma,\alpha} +\mathfrak j_NL_{\gamma,\alpha}}. \] The nesting of the $\mathfrak j_N$ gives transition maps and inverse limits $\widehat Q^{\square,J}$. \begin{theorem}[Two quotient towers] \label{p4:retained:thm:c16-towers} Every verified arrow preserves $\mathfrak j_N$, induces exact isomorphisms in both towers at every stage, and extends bicontinuously to both inverse limits. The raw-to-saturated maps commute with all transitions and arrows. Taking the external orbit direct sum before completion yields restricted orbit--Rees quotients with finite degree support modulo every $\mathfrak j_N$. \end{theorem} \begin{proof} The two towers are formed by imposing the relation lattice before the $\mathfrak j_N$ reduction. Centrality of the scalar ideal makes transported lattice isomorphisms commute with each stage quotient; the restricted orbit--Rees statement is then the inverse-limit form of finite degree support at every precision. The scalar ideal is central, so a transported-lattice isomorphism maps $\mathfrak j_NL$ onto $\mathfrak j_NU_aL$. Combine this with Theorem~\ref{p4:retained:thm:c16-transport} and pass to inverse limits. The restricted support statement is the inverse limit of external direct sums. \end{proof} The whole stage need not have finite rank. Proper-height approximants are finite because only finitely many output colors survive at a fixed precision. \subsection{Quotient-valued finite-stage representatives} Fix one source chamber $\alpha$ with regulated core $\{\mathbb U_{\alpha,\lambda}\}_{\lambda\in X_0}$ and proper height $H_0$. Put $C_N=\{\lambda:H_0(\lambda)