\section{Perturbative, Taylor, and operational boundary layers} \label{p4:ch:lmo-taylor-operational} Fix a marked cobordism and cut it at the boundary where its state is read. Recapping produces a closed composite and a scalar coefficient. Such coefficients can agree across boundary theories even when their state objects differ. We therefore construct the perturbative/Taylor and operational Habiro branches separately and compare them on the geometric closed composites for which the stated theorem supplies a map. This section assembles three boundary theories that have the same closed integer-homology-sphere coefficients but genuinely different state objects. The perturbative layer is the Cheptea--Habiro--Massuyeau (CHM) LMO functor with chamber markings. Its scalar quotient gives a rational Taylor-germ theory at the point $Q=1$. Independently, geometric cups and caps generate an integral Habiro-valued operational envelope. The comparison theorem is coefficientwise on geometric composites; it is not an identification of the two state modules. The earlier lollipop coordinates, rootwise chambers, and quotient towers do not supply maps between these state spaces. The common closed-output test therefore provides the comparison used here, while each boundary state and cotest space retains its own definition. Throughout this section we write $Q$ for the quantum variable used in the Habiro ring and put \[ u=Q-1,\qquad Q=e^h. \] The letter $q$ in CHM's phrase ``$q$-cobordism'' denotes a parenthesization and is unrelated to $Q$. The comparison fixes the same geometric composite in both theories. Chamber markings cancel in its LMO closure, and the closed scalar bridge identifies its perturbative coefficients with the Habiro Taylor series. Geometric sewing in the operational construction reaches the same Habiro element. The boundary quotients retain their separate tests: all diagrammatic cotests on the Taylor side, and geometric cotests on the operational side. Coefficient agreement on the closed composite therefore supplies the stated comparison without identifying those radical quotients. \subsection{The chambered LMO local system} \label{p4:sec:lmo-chambers} Let $F_g$ be the oriented genus-$g$ surface with one boundary component, $H_g=H_1(F_g;\Z)$, and let $L_g\subset H_g$ be the standard primitive integral Lagrangian. Write \[ \Gamma_g=\pi_0\operatorname{Homeo}^+(F_g,\partial F_g),\qquad P_g=\{s\in\Gamma_g:s_*L_g=L_g\}. \] The homological action of $\Gamma_g$ is onto $\operatorname{Sp}(H_g)$ \cite{BCK}, and every primitive integral Lagrangian is the image of $L_g$. For such a Lagrangian $A$, set \[ R_A=\{r\in\Gamma_g:r_*L_g=A\}; \] this is a nonempty right $P_g$-torsor. The normalized CHM functor is \[ \widetilde Z:\mathbf{LCob}_q\longrightarrow{}^{ts}\!\mathcal A, \] where the target is the degree-completed category of top-substantial Jacobi diagrams and composition is contraction of matching colors. In the CHM composition convention, $kh=k\circ h$ and $C(kh)=C(k)\circ C(h)$ for mapping cylinders. The skeletal operator $\widetilde Z(C(s))$ is defined precisely when $s_*L_g=L_g$. Choose markings $r_A\in R_A$, with $r_{L_g}=1$. If $h:A\to hA$ is an arrow in the action groupoid $\operatorname{Lag}_{\Z}(H_g)\rtimes\Gamma_g$, define \begin{equation} K_r(h;A)= \widetilde Z\!\left(C(r_{hA}^{-1}hr_A)\right). \label{p4:eq:lmo-chamber-operator} \end{equation} \begin{theorem}[Chambered LMO transport] \label{p4:thm:lmo-chamber-transport} For every genus, the operators \eqref{p4:eq:lmo-chamber-operator} obey \[ K_r(kh;A)=K_r(k;hA)\circ K_r(h;A). \] If $r'_A=r_As_A$ with $s_A\in P_g$ and $G_A=\widetilde Z(C(s_A))$, then \begin{equation} K_{r'}(h;A)=G_{hA}^{-1}K_r(h;A)G_A. \label{p4:eq:lmo-gauge} \end{equation} Consequently the intrinsic object is a mapping-class-equivariant local system, not a preferred family of matrices in one fixed diagram space. \end{theorem} \begin{proof} The marking \(r_A\) moves each groupoid arrow back to the fixed Lagrangian where the skeletal CHM operator is defined. For a composite arrow, the middle markings cancel, so functoriality gives the stated composition order. Changing the markings inserts a stabilizer operator at each endpoint; these operators appear contragrediently and produce the displayed gauge law. The conjugated word $r_{hA}^{-1}hr_A$ preserves $L_g$, so the CHM domain criterion makes \eqref{p4:eq:lmo-chamber-operator} legal. For composable arrows the two middle markings cancel: \[ r_{khA}^{-1}khr_A =(r_{khA}^{-1}kr_{hA})(r_{hA}^{-1}hr_A). \] The mapping-cylinder homomorphism and CHM functoriality preserve this order. Replacing $r_A$ by $r_As_A$ gives $s_{hA}^{-1}(r_{hA}^{-1}hr_A)s_A$, proving \eqref{p4:eq:lmo-gauge}. Successive changes multiply in the same order, so the gauge maps satisfy the cocycle law. \end{proof} Put \[ V_g={} ^{ts}\!\mathcal A(0,g),\qquad W_g={} ^{ts}\!\mathcal A(g,0). \] The $P_g$-actions are $x\mapsto\widetilde Z(C(s))\circ x$ and $y\mapsto y\circ\widetilde Z(C(s))^{-1}$. Hence \[ \mathscr V_A=R_A\times_{P_g}V_g,\qquad \mathscr W_A=R_A\times_{P_g}W_g \] carry canonical transports $h_*[r,z]=[hr,z]$. Their contraction \[ \langle[r,y],[r,x]\rangle_A=y\circ x \] is well defined because state and cotest gauges cancel contragrediently. No assertion is made that $W_g$ is the full algebraic dual of $V_g$. \subsection{Geometric caps and the closed scalar triangle} \label{p4:sec:lmo-closed-triangle} Let \[ X:0\to(F_g,A),\qquad M:(F_g,A)\to(F_f,B),\qquad Y:(F_f,B)\to0 \] be fixed geometric Lagrangian cobordisms. In chamber markings, their coordinates are obtained by inserting the marking cylinder and its inverse at each internal boundary. \begin{theorem}[Fixed geometric closure] \label{p4:thm:lmo-fixed-closure} The chamber coordinates satisfy \begin{equation} y_t\circ m_{t,r}\circ x_r =\widetilde Z(Y\circ M\circ X) =\Omega\!\left(\widehat{Y\circ M\circ X}\right), \label{p4:eq:lmo-fixed-closure} \end{equation} where the hat denotes oriented recapping and $\Omega$ is the normalized closed LMO invariant. The contraction is independent of $r,t$. With the recursive parenthesization $w_{g+1}=(w_g\,\bullet)$, tensoring the triple by the CHM cup--identity--cap pair leaves the closed value unchanged. \end{theorem} \begin{proof} The inserted marking cylinders cancel inside the composite, leaving the closed geometric cobordism. Recapping identifies its CHM value with the closed invariant, while the three endpoint gauge factors cancel in the contraction. The cup--identity--cap statement is the same cancellation after stabilization. Functoriality cancels the two inserted marking-cylinder pairs before applying $\widetilde Z$, proving the first equality. The composite is a $0\to0$ integral Lagrangian cobordism, hence an integral homology cube; the CHM recapping theorem gives the second equality. Under a marking change the three coordinates acquire respectively $G_A^{-1}$, $G_B^{-1}(-)G_A$, and $(-)G_B$, so every factor cancels. Finally CHM tensor functoriality and $\widetilde Z(\eta)=\widetilde Z(\epsilon)=\varnothing$ prove the explicit canceling-pair statement. \end{proof} A homological Lagrangian does not determine a disk system or a cap. Theorem~\ref{p4:thm:lmo-fixed-closure} is therefore a theorem for fixed geometric morphisms, not for a bare triple of Lagrangian subgroups. Let $\mathcal D=\mathcal A(\varnothing)$ be the completed algebra of closed Jacobi diagrams. Freeze the trace form $\kappa(x,y)=\operatorname{tr}_{\C^2}(xy)$ on $\mathfrak{sl}_2$ and the completed weight system \[ \widehat W_{\mathfrak{sl}_2}^{\mathrm{tr}}:\mathcal D\to\Q[[h]], \qquad \widehat W_{\mathfrak{sl}_2}^{\mathrm{tr}}(\theta)=12h. \] Let \[ \operatorname{Exp}_1:\Q[[u]]\xrightarrow{\sim}\Q[[h]], \qquad f(u)\longmapsto f(e^h-1). \] \begin{theorem}[Closed LMO--Ohtsuki--Habiro bridge] \label{p4:thm:lmo-habiro-closed-bridge} For every oriented integer homology sphere $N$, \begin{equation} \widehat W_{\mathfrak{sl}_2}^{\mathrm{tr}}(\Omega(N)) =\tau^{\mathfrak{sl}_2}(N) =\operatorname{Exp}_1(\tau^O(N)) =\operatorname{Exp}_1(\iota_1(J_N)) \quad\text{in }\Q[[h]]. \label{p4:eq:closed-scalar-triangle} \end{equation} In particular, $\operatorname{Exp}_1^{-1}\widehat W(\Omega(N))=\iota_1(J_N)$ in $\Z[[u]]$. \end{theorem} \begin{proof} The weight system first turns the closed Jacobi-diagram value into the perturbative scalar series. The Kuriya--Lê--Ohtsuki and Habiro comparison results then identify that series in exponential coordinates. Every step is performed after closing the cobordism, which is why the conclusion is a scalar equality and not an inverse statement about boundary diagrams. The trace form has root length squared $2$ and sends the theta graph to $12h$, matching the Kuriya--L\^e--Ohtsuki normalization.\newline Their theorem identifies the weighted LMO invariant with the arithmetic projective $\mathfrak{sl}_2$ perturbative series. On integer homology spheres, at odd levels, the Kirby--Melvin sector and Habiro's normalized WRT invariant give the same scalar. Degreewise uniqueness of the arithmetic expansion then identifies the perturbative series with the exponential-coordinate Ohtsuki series. Habiro's Taylor theorem gives $\tau^O(N)=\iota_1(J_N)$, and his Taylor map is injective. These are equalities of closed scalars; no inverse diagram weight system is used. \end{proof} The first-order checksum is \[ \widehat W(\Omega(N))=1+6\lambda(N)h+O(h^2), \] so changing $Q=e^h$ to $e^{-h}$, reversing the theta cyclic order, or changing the trace normalization would already violate \eqref{p4:eq:closed-scalar-triangle}. \subsection{The rational Taylor boundary functor} \label{p4:sec:taylor-boundary} Set \[ \phi_{\mathrm T}= \operatorname{Exp}_1^{-1}\circ \widehat W_{\mathfrak{sl}_2}^{\mathrm{tr}}: \mathcal D\longrightarrow\Q[[u]]. \] The map is onto: a continuous linear section sends $\sum a_dh^d$ to $\sum a_d12^{-d}\theta^{\sqcup d}$. Thus the scalar quotient is rational, not integral. For the standard genus-$g$ object define \[ \beta_g(y,x)=\phi_{\mathrm T}(y\circ x), \qquad y\in W_g,\ x\in V_g. \] Let $\mathcal R_R^{\mathrm T}(g)$ and $\mathcal R_L^{\mathrm T}(g)$ be the right and left radicals, and put \begin{equation} V_g^{\mathrm T}=V_g/\mathcal R_R^{\mathrm T}(g),\qquad W_g^{\mathrm T}=W_g/\mathcal R_L^{\mathrm T}(g). \label{p4:eq:taylor-radicals} \end{equation} \begin{theorem}[Taylor radical boundary theory] \label{p4:thm:taylor-boundary} The radicals in \eqref{p4:eq:taylor-radicals} are closed. The quotient spaces are complete Hausdorff topological $\Q[[u]]$-modules in their quotient diagram-degree topologies, are $u$-adically separated, and carry a pairing separating in both variables. Every Lagrangian cobordism $C:g\to f$ induces continuous covariant and contravariant maps \[ T_C[x]=[\widetilde Z(C)\circ x],\qquad T_C^\vee[y]=[y\circ\widetilde Z(C)] \] with exact adjoint sewing \[ \overline\beta_f(y,T_Cx)=\overline\beta_g(T_C^\vee y,x). \] These quotients descend to the primitive-Lagrangian chamber local system and obey the gauge law \eqref{p4:eq:lmo-gauge}. \end{theorem} \begin{proof} The radicals are kernels of continuous tests, so they survive passage to the quotient. Diagram contraction preserves them on both sides, and the intersection of all powers of \(u\) removes any residual divisible class. The resulting maps therefore sew adjointly while retaining the chamber gauge action. Each radical is an intersection of kernels of continuous coefficient functionals, hence closed. Diagram spaces are products of finite dimensional homogeneous pieces; quotienting by a closed subspace preserves completeness and Hausdorffness. The central closed-diagram action factors through $\mathcal D/\ker\phi_{\mathrm T}\cong\Q[[u]]$. If a quotient state is divisible by every power of $u$, pairing it against every cotest gives a series in $\bigcap_Nu^N\Q[[u]]=0$; separation then kills the state. Associativity of diagram contraction proves adjoint sewing and preservation of both radicals. The $P_g$-invariance of the radicals allows the same associated-torsor construction as in Theorem~\ref{p4:thm:lmo-chamber-transport}. \end{proof} No perfectness, finite generation, projectivity, or equality of the diagram-degree and $u$-adic topologies is implied. \subsection{The operational Habiro envelope} \label{p4:sec:operational-envelope} Let \[ \mathscr H=\varprojlim_n \Z[Q]/\bigl((Q;Q)_n\bigr),\qquad (Q;Q)_n=\prod_{j=1}^n(1-Q^j), \] and let $\mathfrak I_n$ be the kernel of its $n$th quotient map. For an arbitrary-Lagrangian object $S$, form the restricted completions \begin{align} \widehat{\mathcal P}(S) &=\varprojlim_n\bigoplus_{[X]\in\operatorname{Hom}(0,S)} (\mathscr H/\mathfrak I_n)[X],\notag\\ \widehat{\mathcal P}^{\mathrm{ct}}(S) &=\varprojlim_n\bigoplus_{[Y]\in\operatorname{Hom}(S,0)} (\mathscr H/\mathfrak I_n)[Y]. \label{p4:eq:operational-completions} \end{align} Thus a coefficient family may have growing support, but at every cyclotomic stage its support is finite. These are not unrestricted products over geometric generators. For geometric $X:0\to S$ and $Y:S\to0$, define \begin{equation} \mathcal B_S([Y],[X])=J_{\widehat{Y\circ X}}\in\mathscr H. \label{p4:eq:operational-pairing} \end{equation} Stagewise finite sums extend this uniquely to a continuous pairing on \eqref{p4:eq:operational-completions}. Let $\mathcal R_R(S)$ and $\mathcal R_L(S)$ be its right and left radicals. Quotient by these closed radicals and denote the resulting modules by $\overline{\mathcal P}(S)$ and $\overline{\mathcal P}^{\mathrm{ct}}(S)$. \begin{theorem}[Operational Habiro sewing and universality] \label{p4:thm:operational-habiro} For every $M:S\to T$, geometric composition induces continuous maps \[ M_*[X]=[M\circ X],\qquad M^*[Y]=[Y\circ M] \] on the restricted completions and on their radical quotients. They obey \begin{equation} \mathcal B_T(Y,M_*X)=\mathcal B_S(M^*Y,X). \label{p4:eq:operational-sewing} \end{equation} The quotient pairing is separating; both modules are nonzero, complete Hausdorff cyclotomic modules with faithful $\mathscr H$-action. Moreover, let \[ q_R:\widehat{\mathcal P}(S)\twoheadrightarrow Q_R, \qquad q_L:\widehat{\mathcal P}^{\mathrm{ct}}(S)\twoheadrightarrow Q_L \] be continuous quotient maps to complete Hausdorff $\mathscr H$-modules. If a continuous pairing \[ \gamma:Q_L\times Q_R\longrightarrow\mathscr H \] is separating and factors the completed operational pairing for every $x\in\widehat{\mathcal P}(S)$ and $y\in\widehat{\mathcal P}^{\mathrm{ct}}(S)$, \begin{equation} \mathcal B_S(y,x)=\gamma(q_Ly,q_Rx), \label{p4:eq:operational-universal-factorization} \end{equation} then \[ \ker q_R=\mathcal R_R(S),\qquad \ker q_L=\mathcal R_L(S). \] Hence the operational construction is universal among continuous separated quotient realizations of the already prescribed completed closed pairing. \end{theorem} \begin{proof} Modulo $\mathfrak I_n$ every input is a finite sum, so composition and the pairing are finite and compatible with transition maps. On generators both sides of \eqref{p4:eq:operational-sewing} are literally $J_{\widehat{Y\circ M\circ X}}$; continuity extends the identity. It also shows that both radicals are functorially preserved. Complementary standard cup and cap generators pair to $J_{S^3}=1$, proving nontriviality and faithfulness. For the universal property, if $x\in\ker q_R$, then \eqref{p4:eq:operational-universal-factorization} makes $x$ pair to zero against every cotest, so $x\in\mathcal R_R(S)$. Conversely, if $x\in\mathcal R_R(S)$, surjectivity of $q_L$ makes $q_Rx$ pair to zero against every element of $Q_L$; separation forces $q_Rx=0$. The left-hand argument is symmetric, and the first isomorphism theorem gives uniqueness. \end{proof} \subsection{The exact coefficient comparison} \label{p4:sec:taylor-operational-comparison} \begin{theorem}[Geometric Taylor--Habiro comparison] \label{p4:thm:taylor-habiro-coefficient} For every fixed geometric triple $X,M,Y$, let $N=\widehat{Y\circ M\circ X}$. In any chamber markings, \begin{align} \widehat W(y_t\circ m_{t,r}\circ x_r) &=\operatorname{Exp}_1(\iota_1(J_N)), \label{p4:eq:geometric-h-comparison}\\ \overline\beta_f([y_t],[m_{t,r}\circ x_r]) &=\iota_1(J_N)\in\Z[[u]]\subset\Q[[u]]. \label{p4:eq:geometric-u-comparison} \end{align} At every root of unity $\xi$, \[ \operatorname{ev}_\xi\mathcal B_T([Y],M_*[X]) =\tau_\xi(N). \] The explicit tensor canceling pair preserves $J_N$. \end{theorem} \begin{proof} Evaluate the perturbative and operational constructions on the same fixed geometric composite. The closed-scalar triangle identifies their coefficient at every degree, and the finite-stage sewing maps are compatible with that identification. The comparison is consequently coefficientwise on closed composites; it does not identify the two boundary quotients. The marking cylinders cancel, so the diagrammatic contraction is $\Omega(N)$ by Theorem~\ref{p4:thm:lmo-fixed-closure}. Apply the closed scalar bridge, then $\operatorname{Exp}_1^{-1}$, to obtain the first two identities. Habiro's all-root theorem gives the evaluation identity. For stabilization, the LMO functor sends the added cup and cap to the empty diagram, so the two closed LMO invariants agree. Injectivity of both $\operatorname{Exp}_1$ and Habiro's Taylor map forces equality of the two Habiro elements. \end{proof} \begin{scopefirewall}[The two radical quotients are not identified] \label{p4:firewall:taylor-operational} The Taylor radical is tested by every diagrammatic cotest, whereas the operational radical is tested only by geometric cotests. No density theorem identifies those test families. Therefore Theorem~\ref{p4:thm:taylor-habiro-coefficient} does not descend to a map in either direction between the completed operational quotient and the Taylor radical quotient. It is an exact agreement on their common geometric coefficient domain. \end{scopefirewall} \begin{scopefirewall}[Mandatory perturbative and operational nonclaims] \label{p4:firewall:lmo-operational-nonclaims} Nothing in this section constructs a generic stated-skein state module, a root evaluation of an arbitrary Taylor series, an inverse $\mathfrak{sl}_2$ weight system, a map from arbitrary Jacobi diagrams to the Habiro ring, or an identification with finite RT/GM surface modules. The operational radicals have no intrinsic computable presentation here. No cap is determined by a homological Lagrangian alone. The results do not place the single-$S$ state in the excluded strict $c_0$ carrier, produce a state-level Galois intertwiner, extend to non-IHS coefficients, or define a global Habiro-to-disk map. \end{scopefirewall} \subsection{Primary sources and theorem boundaries} \label{p4:sec:lmo-source-contract} The construction uses \href{https://arxiv.org/abs/math/0701277}{arXiv:\allowbreak math/\allowbreak0701277} for the CHM category, composition order, LMO functor, cups, caps, and recapping; \href{https://arxiv.org/abs/2008.10142}{arXiv:\allowbreak2008.10142} for surjectivity of the bordered symplectic representation; \href{https://arxiv.org/abs/1005.3895}{arXiv:\allowbreak1005.3895} and \href{https://arxiv.org/abs/math/0004099}{arXiv:\allowbreak math/\allowbreak0004099} for the trace-normalized perturbative comparison; and \href{https://arxiv.org/abs/math/0605314}{arXiv:\allowbreak math/\allowbreak0605314} for the Habiro ring, Taylor injection, all-root evaluation, and Ohtsuki equality. The five resulting theorem axes remain separate: Theorems~\ref{p4:thm:lmo-chamber-transport}, \ref{p4:thm:lmo-habiro-closed-bridge}, \ref{p4:thm:taylor-boundary}, \ref{p4:thm:operational-habiro}, and \ref{p4:thm:taylor-habiro-coefficient}.