% Provenance: corrected Hopf-cap/split-completion synthesis, Habiro and % Yang--Mills TeX, Gilmer--Masbaum TeX, and post-consolidation engine audits. \section{Hopf Caps and Split Joint Completions} \label{p4:retained:ch:hopf-cap-split} The joint regulator stores a disk coordinate and a cyclotomic coordinate, but neither coordinate determines the other. This section proves the exact module-level splitting, gives a sharp obstruction to recovering the disk series from Habiro data alone, and constructs the standard complementary-handlebody Hopf cap on a declared endpoint lattice in every finite genus. The cap theorem is intentionally one-sided: it handles the standard filling and its simultaneous transported copies, not arbitrary Heegaard gluings. The section then constructs the rigged genus-one cap pair, proves the contracted stabilization margin, and obtains an all-boundary- color one-way inside-disk endpoint-valuation arrow with a fixed-marking arbitrary-genus tensor extension. Its exact first-cyclotomic obstruction identifies why this still does not give the universal scalar. For the cap application, the module Chinese remainder theorem separates the two directions at each finite stage; its inverse limit gives the joint completion and the finite-stage representative criterion. The cap problem then has its own obstruction: one fixed row cannot encode both complementary solid-torus fillings. The rigged genus-one pair supplies the two contractions, and the stabilization margin controls the genus-normalized contracted kernels. The endpoint-valuation arrow is constructed separately on its one-way disk domain. Its fixed-marking arbitrary-genus tensor extension does not remove the first-cyclotomic obstruction: the raw cutoffs still fail to be cyclotomic Cauchy. \subsection{Finite-stage splitting of the disk-adic and cyclotomic factors} Let \[ \Lambda=\mathbb Z[A],\qquad P_N(A)=\prod_{k=1}^N(1-A^{2k}), \] and define the ideals \[ I_N=(A^{2N}),\qquad C_N=(P_N),\qquad J_N=I_NC_N=(A^{2N}P_N). \] \begin{lemma}[Integral Bezout certificate] \label{p4:retained:lem:c17-bezout} The ideals $I_N$ and $C_N$ are comaximal. If $b_N(A)$ is the truncation through degree $2N-1$ of the formal power series $P_N(A)^{-1}$, then $b_N\in\mathbb Z[A]$ and \[ b_N(A)P_N(A)\equiv1\pmod{A^{2N}}. \] Thus some $a_N\in\mathbb Z[A]$ satisfies \[ a_NA^{2N}+b_NP_N=1. \] \end{lemma} \begin{proof} The constant coefficient of $P_N$ is one, so recursive formal inversion has integral coefficients. Truncating through degree $2N-1$ gives the congruence, and its difference from one is divisible by $A^{2N}$. \end{proof} Set \[ e_N^{\rm disk}=b_NP_N, \qquad e_N^{\rm cyc}=1-e_N^{\rm disk}. \] Modulo $J_N$, these are complementary orthogonal idempotents: the first is $1$ modulo $A^{2N}$ and $0$ modulo $P_N$, while the second has the opposite residues. \begin{theorem}[Module Chinese remainder theorem] \label{p4:retained:thm:c17-module-crt} For every $\Lambda$-module $M$, without any flatness or finiteness hypothesis, \[ M/J_NM\xrightarrow{\sim} M/A^{2N}M\times M/P_NM, \qquad [x]\longmapsto([x],[x]). \] An inverse sends $([x],[y])$ to \[ [e_N^{\rm disk}x+e_N^{\rm cyc}y]_{J_N}. \] \end{theorem} \begin{proof} The two ideals are separated by a finite Bezout identity: the idempotents produce the two prescribed residues, while the same identity sends an element lying in both submodules into their product. This proves the module statement directly and explains why no flatness or finiteness hypothesis is needed. The idempotents prove surjectivity. If $x\in A^{2N}M\cap P_NM$, the Bezout identity gives \[ x=(a_NA^{2N}+b_NP_N)x. \] Because $x$ belongs to both submodules, both summands are in $A^{2N}P_NM$. Hence \[ A^{2N}M\cap P_NM=A^{2N}P_NM=J_NM, \] which proves injectivity. \end{proof} \paragraph{Worked checkpoint.} At joint stage $N=2$, \[ P_2=1-A^2-A^4+A^6, \qquad J_2=A^4-A^6-A^8+A^{10}. \] The Bezout pair $a_2=2-A^4$, $b_2=1+A^2$ satisfies \[ (2-A^4)A^4+(1+A^2)P_2=1. \] Hence \[ e_2^{\rm disk}=1-2A^4+A^8, \qquad e_2^{\rm cyc}=2A^4-A^8. \] Their idempotency defects, their product, and $e_2^{\rm disk}+e_2^{\rm cyc}-1$ all vanish modulo $J_2$. Apply this theorem to either raw or saturated quotient module of Section~\ref{p4:retained:ch:raw-saturated}. It gives, naturally under every verified diagram-labeled transport, \[ \frac{L}{\mathscr X+J_NL} \cong \frac{L}{\mathscr X+A^{2N}L} \times \frac{L}{\mathscr X+P_NL}, \] where $\mathscr X$ is either complete relation lattice. \subsection{The split completion and finite-stage representatives} The finite-stage splittings commute with the transition maps. \begin{theorem}[Split joint completion] \label{p4:retained:thm:c17-split} For every $\Lambda$-module $M$ there is a canonical natural isomorphism \[ \widehat M_{\rm joint}:=\varprojlim_NM/J_NM \cong \widehat M_{\rm disk}\times\widehat M_{\rm cyc}, \] where \[ \widehat M_{\rm disk}=\varprojlim_NM/A^{2N}M, \qquad \widehat M_{\rm cyc}=\varprojlim_NM/P_NM. \] For $M=\Lambda$, the disk factor is $\mathbb Z[[A]]$ and the cyclotomic factor is the $A^2$-Habiro-type completion $\varprojlim_N\mathbb Z[A]/(A^2;A^2)_N$. \end{theorem} \begin{proof} The split completion is obtained in two layers. First the finite-stage Chinese remainder theorem gives compatible disk and cyclotomic coordinates; then inverse limits turn those compatible splittings into the joint completed module. The finite-stage representative criterion is checked against both filtrations, which is why it does not collapse the two completion directions into one. Take the inverse limit of Theorem~\ref{p4:retained:thm:c17-module-crt}. Finite products commute with inverse limits. The even powers $(A^{2N})$ are cofinal with the ordinary $A$-adic filtration. \end{proof} \begin{theorem}[Criterion for finite-stage representatives in the split completion] \label{p4:retained:thm:c17-common-criterion} For a sequence $f_N\in M$, the following are equivalent: \begin{enumerate} \item $f_{N+1}-f_N\in J_NM$ for every $N$; \item simultaneously \[ f_{N+1}-f_N\in A^{2N}M, \qquad f_{N+1}-f_N\in P_NM \] for every $N$. \end{enumerate} Such a sequence determines one pair in $\widehat M_{\rm disk}\times\widehat M_{\rm cyc}$. Conversely, every pair has compatible joint residues reconstructed by the CRT idempotents. \end{theorem} \begin{proof} The criterion is the module intersection identity behind the finite CRT: the joint regulator is exactly the intersection of the disk and cyclotomic regulators. Passing to inverse limits then records the two compatible coordinates, while the finite idempotents reconstruct a joint residue from any such pair. The equivalence is exactly $A^{2N}M\cap P_NM=J_NM$ from Theorem~\ref{p4:retained:thm:c17-module-crt}. The inverse-limit statement is Theorem~\ref{p4:retained:thm:c17-split}. \end{proof} Thus a proper-height construction is ``common'' because the same finite polynomials converge in both topologies. The two limits remain independent coordinates. \subsection{No continuous Habiro-to-disk identity map} \begin{theorem}[Cyclotomic-to-disk obstruction] \label{p4:retained:thm:c17-no-map} There is no continuous additive map \[ \Psi:\widehat\Lambda_{\rm cyc}\longrightarrow\mathbb Z[[A]] \] whose restriction to $\Lambda$ is ordinary polynomial inclusion. In particular, there is no continuous ring homomorphism with that property. \end{theorem} \begin{proof} The obstruction compares the two topologies on the same polynomial sequence. The cyclotomic filtration makes the sequence tend to zero because fixed factors eventually divide it, while its constant coefficient keeps its disk order at zero. A continuous extension of polynomial inclusion would have to preserve both behaviors, giving the contradiction. The polynomial sequence $P_N=(A^2;A^2)_N$ tends to zero in the cyclotomic topology because every fixed $P_m$ divides $P_N$ for $N\geq m$. If $\Psi$ extended the identity, then $\Psi(P_N)=P_N$. But each $P_N$ has constant coefficient one, hence $A$-order zero, and does not even tend to zero modulo $A$. This contradicts continuity at zero. \end{proof} The disk factor in $J_N$ is therefore genuine extra data. A closed unified Habiro invariant can supply a cyclotomic coordinate, but not a canonical disk series by continuous polynomial extension. When an identity such as a false-theta formula presents both coordinates, it is a chosen compatible disk--cyclotomic lift, not a consequence of the closed cyclotomic coordinate alone. \subsection{The standard arbitrary-genus Hopf cap} Work in the closed, unpointed handlebody sector. Let $g\geq1$, let $H_g\subset S^3$ be the standard unknotted genus-$g$ handlebody and $H'_g$ its complement. Define the Laurent skein cap \[ \epsilon_H:\operatorname{Sk}_A(H_g;R)\longrightarrow R, \qquad \epsilon_H(x)=\langle x\cup\varnothing\rangle_{S^3}, \qquad R=\mathbb Z[A,A^{-1}], \] and extend it $\FA$-linearly to the ambient generic space. Let $\mathcal D_0$ be an ordered complete disk system of $H'_g$; filling along it and adding the final three-handle gives $S^3$. For $g=0$ the statement is tautological: take $H_0=B^3$, $L_H=L_\star=\Lambda\mathbf1$, and $c_0=1$. Choose any endpoint sign $\varepsilon\in\{-,+\}^g$ and its lattice \[ L_H^\varepsilon= \bigoplus_\lambda\Lambda\mathsf e_\lambda^\varepsilon. \] At the all-zero coloring $\lambda_0$, every endpoint scalar equals one, so $\mathsf e_{\lambda_0}^\varepsilon=1$. The universal core basis is \[ \mathbb U_\lambda= D_\star(\lambda) A^{\sum_{i=1}^g2(a_i+b_i+1)^2} \prod_{i=1}^gp_{b_i}(A)\widetilde b_\lambda. \] Put \[ c_g=D_\star(\lambda_0)A^{2g}\in\Lambda\setminus\{0\}. \] \begin{theorem}[Standard Hopf-cap theorem] \label{p4:retained:thm:c17-hopf-cap} For every finite $g\geq1$: \begin{enumerate} \item the cap annihilates the complete successive-slide module of the standard complementary filling, \[ \epsilon_H(\mathcal J_{\mathcal D_0}^R)=0; \] \item on every declared endpoint lattice it is the primitive $\Lambda$-valued row \[ \epsilon_H(\mathsf e_\lambda^\varepsilon) =\delta_{\lambda,\lambda_0}; \] \item on the universal core it is \[ \epsilon_H(\mathbb U_\lambda) =\delta_{\lambda,\lambda_0}c_g; \] \item taking $L_H^\varepsilon$ as the source chamber, the cap annihilates both the raw and saturated relation lattices and descends at every joint stage; and \item on a regulated finite core vector $v_N=\sum_\lambda x_{N,\lambda}\mathbb U_\lambda$, \[ \epsilon_H(v_N)=c_gx_{N,\lambda_0}. \] Hence a joint-Cauchy vector net gives a joint-Cauchy scalar net. \end{enumerate} \end{theorem} \begin{proof} The complementary filling identifies the full successive-slide kernel, while the Hopf pairing isolates the all-zero coloring as the vacuum row. Restricting that row to the declared endpoint lattice makes it integral and allows it to descend through both relation lattices; the universal-core formula then records the same scalar at finite joint stages. The theorem remains tied to this declared filling and lattice. The successive two-handle theorem \cite{Przytycki} identifies the kernel of \[ \operatorname{Sk}_A(H_g;R)\longrightarrow \operatorname{Sk}_A(S^3;R)=R \] with the full successive-slide module, proving the first claim. The lollipop vectors are orthogonal for the complementary-handlebody Hopf pairing, and the empty complementary vector is the all-zero vector. Pairing with it therefore kills every $\widetilde b_\lambda$ except $\lambda_0$. Endpoint normalization is one there, proving the primitive row. On the core, all $p_{b_i}$ equal one at the vacuum and the displayed monomial is $A^{2g}$, giving $c_g$; the lollipop pairing normalization is matched to \cite{GilmerMasbaum}. The endpoint raw relation lattice lies in the full slide module. The $\FA$-linear extension of $\epsilon_H$ kills its $\FA$-span and hence the saturated intersection. Because the endpoint row is $\Lambda$-valued, it reduces modulo every $J_N$. The final scalar formula follows immediately. \end{proof} \paragraph{Exact scope.} The theorem is an integral cotest theorem on the declared endpoint lattice and its universal-core sublattice. It is not a claim that $\epsilon_H$ is $\Lambda$-valued on the whole Laurent skein module. For the solid-torus core $z$, \[ \epsilon_H(z)=-(A^2+A^{-2}). \] It also does not compare two endpoint source chambers chosen independently. \subsection{Transport covariance is not a Heegaard amplitude} For a verified diagram-labeled arrow $\gamma$, set \[ L_\gamma^H=U_\gamma L_H^\varepsilon,\qquad L_{\gamma,\star}=U_\gamma L_\star,\qquad \epsilon_\gamma=\epsilon_H\circ U_\gamma^{-1}, \] and carry the disk system to $\gamma\mathcal D_0$. \begin{theorem}[Cap local system on the verified orbit] \label{p4:retained:thm:c17-cap-local} The row $\epsilon_\gamma$ is $\Lambda$-valued on $L_\gamma^H$, kills the transported raw and saturated relation lattices, and satisfies \[ \epsilon_\gamma(U_\gamma\mathsf e_\lambda^\varepsilon) =\delta_{\lambda,\lambda_0}, \qquad \epsilon_\gamma(U_\gamma\mathbb U_\lambda) =\delta_{\lambda,\lambda_0}c_g. \] Every stage scalar is invariant under the exact relations already imposed in the verified orbit. Unidentified twist degrees remain distinct. \end{theorem} \begin{proof} The cap local system is transported rather than recomputed at each endpoint: conjugate the fixed cap row by the inverse transport, carry the marking and disk system with it, and use exact lattice and relation transport. Evaluation on a transported basis then reduces to the base cap, while the later mixed amplitude keeps one handlebody datum fixed and is therefore a different operation. Apply $\epsilon_H$ after $U_\gamma^{-1}$ and use exact simultaneous transport of the lattice, diagram, and relation module. \end{proof} For every source vector $v$, \[ \epsilon_\gamma(U_\gamma v)=\epsilon_H(v). \] This identity is coordinate covariance, not a root-specialized mixed amplitude. Transporting the state and cap contragrediently cancels the gluing operator. A Heegaard amplitude instead keeps one handlebody datum fixed and takes a mixed coefficient such as \[ \epsilon_H(U_\phi v_0). \] \begin{proposition}[Genus-one fixed-cap obstruction] \label{p4:retained:prop:c17-two-fills} No nonzero $\FA$-linear row on the solid-torus skein module annihilates both the complete longitude-filling and meridian-filling ideals. Thus one normalized row cannot be both the standard $S^3$ Hopf cap and the $S^1\times S^2$ identity-double cap. \end{proposition} \begin{proof} The $S^3$ cap has \[ \epsilon_H(1)=1,\qquad \epsilon_H(z)=-(A^2+A^{-2}), \] whereas the normalized identity-double row \cite{YangMills} has \[ \operatorname{YM}(1)=1,\qquad \operatorname{YM}(z)=0. \] Each filled generic skein quotient is one-dimensional, so its annihilator is a one-dimensional row line. The normalized generators differ on $z$; the lines are distinct and intersect only in zero. \end{proof} This is the sharp separator: re-expressing one cap with its carried diagram is proved, but arbitrary fillings require a coherent family indexed by handlebody double cosets. \subsection{The genus-one rigged two-cap pair} \label{p4:retained:sec:c17-rigged-two-cap} The preceding obstruction forbids one fixed row, not a compatible pair of rows on a rigged primal--cotest system. Habiro's diagonal annular bases provide such a pair at genus one \cite{Habiro}. Use his variables \(v=e^{h/2}\), \(q_H=v^2\), and translate to the chamber convention by \[ v=A^2,\qquad q_H=A^4=q^2, \qquad q=A^2. \] The braided dictionary also rephases colors: \[ V_i^{\rm H}\longmapsto(-1)^if_i^{\rm \FA}, \qquad V_1^{\rm H}\longmapsto -V_1. \] Indeed, the Kauffman Hopf matrix is \(\Pi_{ij}=(-1)^{i+j}[(i+1)(j+1)]_{A^2}\) \cite{Marche}, while Habiro's is \([(i+1)(j+1)]_v\). The two signs from the rephased colors cancel those in \(\Pi\). This full dictionary is forced: the scalar shortcut \(v=-A^2\) matches the first quantum dimension but gives the wrong odd--odd Hopf entries. Write \(V_i\) for the irreducible color basis, so \(V_i=U_i(V_1/2)\). Habiro's Newton and even families are \[ P_n=\prod_{i=0}^{n-1} (V_1-v^{2i+1}-v^{-2i-1}), \qquad S_n=\prod_{i=1}^n (V_1^2-(v^i+v^{-i})^2). \] His exact Hopf diagonal is \begin{equation} \langle P_m,S_n\rangle_H =\delta_{mn}\prod_{j=2}^{2n+1}(v^j-v^{-j}). \label{p4:retained:eq:c17-habiro-diagonal} \end{equation} The identity-double Yang--Mills row is the coefficient of \(V_0\), \(\operatorname{YM}(V_i)=\delta_{i0}\) \cite{YangMills}. Habiro's basis-change formula therefore gives \begin{equation} h_n(v):=\operatorname{YM}(P_n) =(-1)^n\frac{[2]}{[n+2]} \begin{bmatrix}2n+1\\n+1\end{bmatrix}_{\!v} =(-1)^nv^{-n^2}\mathcal C_n(v^2), \label{p4:retained:eq:c17-YM-Pn} \end{equation} where \begin{equation} \mathcal C_n(q)= \frac{1-q^2}{1-q^{n+2}} \begin{bmatrix}2n+1\\n+1\end{bmatrix}_{\!q} \in\mathbb Z[q]. \label{p4:retained:eq:c17-Cn} \end{equation} Indeed, if \(d\mid n+2\) and \(d\geq3\), the exponent of \(\Phi_d(q)\) in the Gaussian polynomial in \eqref{p4:retained:eq:c17-Cn} is \[ \left\lfloor\frac{2n+1}{d}\right\rfloor -\left\lfloor\frac{n+1}{d}\right\rfloor -\left\lfloor\frac n d\right\rfloor=1. \] For \(d=1,2\), the numerator \(1-q^2\) supplies the required factor. Thus the quotient is polynomial; it has constant coefficient one, and \[ \mathcal C_n(1)=\frac{2}{n+2}\binom{2n+1}{n+1} =\operatorname{Cat}_{n+1}. \] The raw master regulator does not by itself clear this row: \[ \ord_A\bigl(J_nh_n(A^2)\bigr)=2n-2n^2, \] which is negative for \(n\geq2\). The required polynomial primal and cotest families are instead \begin{align*} \mathsf P_n &=(-1)^nA^{2n^2} P_n\big|_{v=A^2,\,V_1^{\rm H}=-V_1} =\prod_{i=0}^{n-1} (A^{4i+2}V_1+A^{8i+4}+1),\\ \mathsf S_n &=A^{2n^2+8n} S_n\big|_{v=A^2,\,V_1^{\rm H}=-V_1}\\ &=A^{6n}\prod_{i=1}^n \bigl(A^{4i}V_1^2-(A^{4i}+1)^2\bigr). \end{align*} Put \[ D_n=A^{2n}\prod_{j=2}^{2n+1}(1-A^{4j}). \] Then \begin{equation} \langle\mathsf P_m,\mathsf S_n\rangle_{\rm \FA} =\delta_{mn}(-1)^nD_n, \qquad D_n\in J_n\Lambda, \label{p4:retained:eq:c17-cleared-Hopf-diagonal} \end{equation} For \(n\geq1\), the divisibility can be seen without localization from the explicit quotient \[ \frac{D_n}{J_n} =\frac{1-A^{4n+4}}{1-A^2} \prod_{k=2}^{n}(1+A^{2k}) \prod_{j=n+2}^{2n+1}(1-A^{4j})\in\Lambda, \] with empty products understood. Hence \(D_n\in J_n\Lambda\), and the chosen power \(A^{2n^2+8n}\) is the smallest polynomial clearing of \(S_n\) for which the paired diagonal has the master regulator's disk order \(2n\). Indeed, raw \(S_n\) has the unique lowest order \(-2n(n+1)\), while, after the fixed \(A^{2n^2}\) primal clearing, an \(A^sS_n\) cotest clearing gives paired order \(s-2n^2-6n\). Requiring this to be at least \(2n\) forces \(s\geq2n^2+8n\), with equality above. Define the fractional cotest generator over \(\FA\) by \[ \mathsf S_n^\dagger=D_n^{-1}\mathsf S_n. \] This is a declaration of a lattice in the generic dual; it does not invert \(D_n\) inside \(\widehat\Lambda_J\). Let the ambient restricted primal module be \[ \mathscr P^{\rm res}= \left\{\begin{gathered}\sum_{n\geq0}a_n\mathsf P_n: a_n\in\widehat\Lambda_J,\ \\ \text{for every \(N\), all but finitely many \(a_n\) lie in } J_N\widehat\Lambda_J\end{gathered}\right\}. \] Equivalently, this is the inverse limit of the finite-support coordinate modules modulo \(J_N\). It contains both the algebraic lattice \(L_P=\bigoplus_n\Lambda\mathsf P_n\) and the stricter coercive module \[ \mathscr P^{\rm coer}= \left\{\sum_{n\geq0}J_nx_n\mathsf P_n: x_n\in\widehat\Lambda_J\right\}\subset\mathscr P^{\rm res}. \] On the cotest side put \[ \mathscr S^\dagger= \prod_{n\geq0}\widehat\Lambda_J\mathsf S_n^\dagger. \] Give \(\mathscr P^{\rm res}\) its inverse-limit restricted-product topology and \(\mathscr S^\dagger\) the coefficientwise product topology. \begin{theorem}[Rigged genus-one two-cap theorem] \label{p4:retained:thm:c17-rigged-two-cap} The formula \begin{equation} \left\langle \sum_na_n\mathsf P_n, \sum_nb_n\mathsf S_n^\dagger \right\rangle_{\rm rig} =\sum_{n\geq0}(-1)^na_nb_n \label{p4:retained:eq:c17-rigged-pairing} \end{equation} defines a continuous \(\widehat\Lambda_J\)-valued pairing \(\mathscr P^{\rm res}\times\mathscr S^\dagger\to\widehat\Lambda_J\). The two cotests \[ Y_{S^3}=\mathsf S_0^\dagger, \qquad Y_{\rm id}=\sum_{n\geq0} (-1)^n\mathcal C_n(A^4)\mathsf S_n^\dagger \] represent the standard \(S^3\) and identity-double cap rows. Explicitly, \begin{align*} \epsilon_{S^3}\left(\sum_na_n\mathsf P_n\right)&=a_0,\\ \epsilon_{\rm id}\left(\sum_na_n\mathsf P_n\right) &=\sum_{n\geq0}\mathcal C_n(A^4)a_n. \end{align*} On \(\mathscr P^{\rm coer}\), these become \[ \epsilon_{S^3}=x_0,\qquad \epsilon_{\rm id}=\sum_{n\geq0}J_n\mathcal C_n(A^4)x_n. \] For polynomial coordinates \(x_n\in\Lambda\), their finite partial sums are literal common disk--cyclotomic polynomial approximants. For general \(x_n\in\widehat\Lambda_J\), the partial sums are joint-Cauchy truncations; polynomial approximants require a compatible choice of representatives at each finite level. On the algebraic primal lattice \(L_P\subset\mathscr P^{\rm res}\), the rows have the correctly typed filling descent \begin{align*} \epsilon_{S^3} (L_P\cap\mathcal J_{\rm longitude}^{R})&=0,\\ \epsilon_{\rm id} (L_P\cap\mathcal J_{\rm meridian}^{R})&=0. \end{align*} They also kill the corresponding saturations inside \(L_P\), and, by continuity, the closures in \(\mathscr P^{\rm res}\) of the two displayed intersections. \end{theorem} \begin{proof} The cleared diagonal pairing turns the primal and fractional cotest families into a coefficientwise rigged pairing. Restricted support makes the pairing finite at every joint stage, while the two selected cotests recover the standard \(S^3\) and identity-double rows separately. Filling descent is then inherited from the corresponding relation annihilators; the construction does not identify the two rows. Equation~\eqref{p4:retained:eq:c17-cleared-Hopf-diagonal} gives \eqref{p4:retained:eq:c17-rigged-pairing}. Modulo \(J_N\), the definition of \(\mathscr P^{\rm res}\) leaves only finitely many primal coordinates. This proves convergence and continuity; on \(\mathscr P^{\rm coer}\), every term with \(n\geq N\) vanishes. Pairing with \(Y_{S^3}\) selects the zeroth coefficient. Pairing with \(Y_{\rm id}\) cancels the two signs in the diagonal and gives \(\operatorname{YM}(\mathsf P_n) =\mathcal C_n(A^4)\), proving the displayed rows and their common-tail property. Moreover, \[ \mathsf P_1=A^2V_1+A^4+1=r_\ell, \] so the fixed Kauffman \(S^3\) cap kills \(\mathsf P_n\) for every \(n\geq1\), while it sends \(\mathsf P_0\) to one. This identifies the coefficient selector with the actual cap, rather than merely a formally similar row. The first row is the standard complementary-handlebody cap, while the second is inclusion into the identity double followed by the Yang--Mills coefficient functional. The successive two-handle theorem therefore annihilates the appropriate full Laurent slide module before restriction to \(L_P\). Corollary~\ref{p4:retained:cor:c16-integral-cap} gives saturated descent, and continuity gives the stated closures inside \(\mathscr P^{\rm res}\). \end{proof} The rows remain genuinely distinct. For the cleared longitude relation \[ r_\ell=A^2V_1+(A^4+1)1, \] one has \[ \epsilon_{S^3}(r_\ell)=0, \qquad \epsilon_{\rm id}(r_\ell)=A^4+1\neq0. \] Thus the theorem constructs two filling-specific row lines; it does not evade Proposition~\ref{p4:retained:prop:c17-two-fills}. There is also a basis-independent obstruction to turning the generic color Hopf kernel into a bare integral \(S\)-isomorphism. Let \[ H_r(v)=([ab]_v)_{1\leq a,b\leq r}. \] \begin{proposition}[Unbounded integral Hopf index] \label{p4:retained:prop:c17-Hopf-index} One has the exact determinant \begin{equation} \det H_r(v)= \prod_{j=1}^r[j]_v \prod_{1\leq i0$, the positive endpoint has minimum $5$ and the negative endpoint has minimum $8$. For an active standard handle put $c=a$, $n=m+d$, $d\geq0$. The exact pole orders of the full positive and negative contracted local kernels are \begin{align*} P_+(c,n,m)&=(n+c)^2+n+m+4c,\\ P_-(c,n,m)&=n^2+(4c+2m+1)n+3c^2+2cm+5c+2m +3(n+c). \end{align*} The first is read from the terminating product in Theorem~\ref{p4:retained:thm:c12-kernel}; for the second, the mirror formula has a unique lowest term. With $R_c(s)=2(c+s+1)^2$, the two shifted margins are \begin{align*} R_c(n)+R_c(m)-P_+ &=4+4c+3c^2+6m+6cm+3m^2 +3d+2cd+2md+d^2,\\ R_c(n)+R_c(m)-P_- &=4+(c+m)^2+2m+d^2. \end{align*} They are at least $4$, with equality only for $c=n=m=0$. The passage from the subset-specific scalar vector to the full universal core adds the nonnegative active order \(n(n+1)/2+m(m+1)/2\), so these are conservative lower bounds for the full core kernel. An inactive handle contributes $R_{a}(b)+R_{a'}(b')\geq4$ through the two factors $E_S$. All remaining denominator-clearing and spectator factors in Theorem~\ref{p4:retained:thm:c13-universal-core} are polynomials, so they cannot lower the $A$-order. Summing the handlewise bounds proves divisibility by $A^{4g}$ for every nonempty square-free standard family. For the all-negative standard multitwist, orthogonality and triangularity give on the diagonal \[ \kappa^{\star,-[g]}_{\lambda\lambda} =\left(\prod_{i=1}^g\mu_{a_i+b_i}^{-1}\right) \varrho_\lambda^\star. \] Multiplying by the polynomial monomial $\prod_i\mu_{a_i+b_i}$ proves $\varrho_\lambda^\star\in A^{4g}\Lambda$. The sep\-arating-edge formula adds only \[ \sum_{e\in E}(1+\delta_e)j_e(j_e+2)\geq0 \] to this order, and the combined formula adds the same nonnegative monomial to an already divisible standard kernel. This proves the third claim. If all handle pairs are $(a_i,b_i)=(0,0)$, admissibility forces every tree color to vanish: remove the lollipop loops and induct from a cherry of the remaining rooted tree, where a triple $(0,0,x)$ forces $x=0$. Hence the global equality case is the unique vacuum. Regauging both inputs multiplies a contracted kernel by $A^{-4g}$, so the preceding divisibility proves polynomiality. Finally $\overline{\mathbb U}_{\lambda_0,g}=1$; the empty skein has pairing one and is fixed by every displayed twist and by empty-graph stabilization. \end{proof} \paragraph{Scope.} The theorem closes the contracted stabilization-margin question for the operator families already constructed: standard square-free twists, separating square-free twists, and their combined contracted kernels. It does not construct the generic Hopf/$S$ row, prove a complete mixed mapping-class presentation, or supply the Maslov and normalization data required by Theorem~\ref{p4:retained:thm:c07-conditional-descent}. Nor does it yet prove a genus-$g$ to genus-$(g+1)$ lattice embedding away from the vacuum; that would require compatibility of the nonvacuum denominator-clearing factor systems. Those inputs must be checked on the same normalized core when they are constructed. \subsection{Relative Newton duals and the one-holed-torus disk \texorpdfstring{\(S\)}{S} arrow} \label{p4:retained:sec:c17-relative-disk-S-arrow} This section isolates a construction which is stronger than the two-cap pairing but weaker than the missing generic mapping-class operator. For every boundary half-color \(c\geq0\), the relative Newton basis admits a polynomial rigging with a jointly restricted Hopf diagonal. In the fixed Gilmer--Masbaum coordinates, the finite word \(t t^\star t\) then has a coefficientwise \(A\)-adic limit after the exact boundary-dependent input shift computed below. The resulting map is a one-way inside-disk valuation arrow. Its raw cutoffs fail even the first cyclotomic congruence, so it is neither an invertible generic \(S\)-operator nor a root base change. \paragraph{Convention boundary.} The relative lollipop formulas use the Gilmer--Masbaum variable \(q=A^2\). For the uncolored Habiro pair, the full braided dictionary is \[ v=A^2,\qquad q_H=v^2=A^4,\qquad V_i^{\rm H}\longmapsto(-1)^if_i^{\rm \FA}. \] In particular \(V_1^{\rm H}\mapsto -V_1^{\rm \FA}\). The scalar shortcut \(v=-A^2\) happens to reproduce the Newton polynomials but gives the wrong odd--odd Hopf entries; it is not a braided dictionary. \subsubsection{The all-color relative Newton rigging} Recall \[ Q_n^{(c)} =L_{c,0}\prod_{i=c}^{c+n-1}(z-\lambda_i), \qquad \lambda_i=-q^{i+1}-q^{-i-1}, \] and put \[ D(c,n)=(c+n)^2+3c+2n,\qquad J_n=q^n(q;q)_n. \] Define \[ u_{c,n}=n(n+2c+1),\qquad v_{c,n}=n^2+2cn+5n+2c^2+6c. \] Then \[ u_{c,n}+v_{c,n}=2\bigl(D(c,n)+n\bigr). \] Set \begin{align*} \mathsf P_{c,n} &=A^{u_{c,n}}Q_n^{(c)}\\ &=L_{c,0}\prod_{r=1}^{n} \bigl(1+A^{2(c+r)}z+A^{4(c+r)}\bigr),\\ \mathsf R_{c,n} &=A^{v_{c,n}}Q_n^{(c)} =A^{2c^2+6c+4n}\mathsf P_{c,n}. \end{align*} The letters \(\mathsf P,\mathsf R\) avoid confusing this self-dual relative Newton system with Habiro's distinct \(P_n,S_n\) pair. Habiro's \(P_n\) samples the even-index eigenvalue nodes, whereas \(Q_n^{(c)}\) uses the consecutive nodes \(\lambda_c,\ldots,\lambda_{c+n-1}\). \begin{theorem}[Relative Newton rigging] \label{p4:retained:thm:c17-relative-Newton-rigging} For every \(c,n\geq0\), both displayed families are polynomial relative to the fixed graph \(L_{c,0}\), span the same \(\FA\)-space, and satisfy \[ B_c(\mathsf P_{c,m},\mathsf R_{c,n}) =\delta_{mn}\Delta_{c,n}, \] where \[ \Delta_{c,n} =q^{D(c,n)+n}\eta_{c,n} =(-1)^cq^n \frac{(q;q)_n(q;q^2)_{c+n+1}(-q;q)_{2c+n+1}} {(1-q^2)\genfrac{[}{]}{0pt}{}{2c}{c}_{q^2}} \in J_n\mathbb Z[q]. \] Consequently \[ \mathsf R_{c,n}^{\dagger} =\Delta_{c,n}^{-1}\mathsf R_{c,n} \] defines a restricted fractional dual and \[ \left\langle \sum_{n\geq0}J_nx_n\mathsf P_{c,n}, \sum_{n\geq0}b_n\mathsf R_{c,n}^{\dagger} \right\rangle =\sum_{n\geq0}J_nx_nb_n \] is a continuous joint pairing. At every joint stage only finitely many coordinates survive. \end{theorem} \begin{proof} Relative Newton factors are first cleared into two polynomial families, after which orthogonality supplies the diagonal norm. The exact norm divisibility provides the joint regulator needed for the fractional dual, and the stagewise vanishing of higher colors gives continuity. This is a topology-aware dual construction, not a unimodular state-to-state operator. Multiplying the \(r\)-th Newton factor by \(A^{2(c+r)}\) gives the first polynomial product, so \(u_{c,n}\) is its exact minimal monomial clearing. The difference \[ v_{c,n}-u_{c,n}=2c^2+6c+4n \] is nonnegative, giving the second polynomial family. Orthogonality of \(Q_n^{(c)}\), the exponent identity, and the exact relative norm factorization give the displayed diagonal. Its divisibility by \(J_n\) is the cyclotomic floor estimate in Theorem~\ref{p4:retained:thm:c11-norm}. Modulo \(J_N\), every term with \(n\geq N\) vanishes. \end{proof} For \(c>0\), the vacuum root norm satisfies \(\eta'_{c,0}\sim h^c\). Thus \(\mathsf R_{c,0}^{\dagger}\) belongs to the restricted fractional dual, not the original integral root lattice. Relative duality is therefore not a unimodular state-to-state \(S\). \subsubsection{The finite cutoff word} Use \[ Q_n^{\prime(c)}=\frac{Q_n^{(c)}}{\{n\}!}, \qquad \eta'_{c,n}=B_c(Q_n^{\prime(c)},Q_n^{\prime(c)}). \] Let \(b^{(c)}_{nm}\), \(n\geq m\), be the lower-triangular matrix of the positive longitude twist \(t^\star\). Hopf adjointness defines the upper-triangular meridian matrix \[ a^{(c)}_{mn} =\frac{\eta'_{c,n}}{\eta'_{c,m}}b^{(c)}_{nm}, \qquad m\leq n. \] Let \(A_{c,N}\) and \(B_{c,N}\) be their leading blocks through color \(N\), and set \[ \mathsf S_{c,N}=A_{c,N}B_{c,N}A_{c,N}. \] At an admissible odd prime \(p\), with \(d=(p-1)/2\) and \(N=d-c-1\), this is the finite Gilmer--Masbaum matrix of \(t t^\star t\). In the generic calculation \(N\) is retained as a cutoff. The exact orders are \begin{align*} \ord_A\eta'_{c,n} &=-2c^2-6c-4cn-n^2-3n,\\ \ord_A b^{(c)}_{nm} &=\frac{2c^2+4c-m+m^2+5n+4cn+n^2}{2},\\ \ord_A a^{(c)}_{mn} &=\frac{2c^2+4c+5m+8cm+3m^2-n-4cn-n^2}{2}. \end{align*} \begin{theorem}[All-color inside-disk valuation arrow] \label{p4:retained:thm:c17-relative-disk-S-arrow} For every \(c,r,m\geq0\), the coefficientwise limit \[ s^{\rm in}_{c;rm} =\lim_{N\to\infty}(\mathsf S_{c,N})_{rm} \in\mathbb Q((A)) \] exists in the \(A\)-adic topology. Its exact order is \begin{equation} \ord_A s^{\rm in}_{c;rm} =3c(c+2) +\frac{r(3r+8c+9)-m(m+4c+1)}2. \label{p4:retained:eq:c17-relative-S-entry-order} \end{equation} For \(N>\max\{r,m\}\), \begin{equation} (\mathsf S_{c,N}-\mathsf S_{c,N-1})_{rm} =a^{(c)}_{rN}\sum_{j=0}^{m}b^{(c)}_{Nj}a^{(c)}_{jm}, \label{p4:retained:eq:c17-relative-S-cutoff-difference} \end{equation} and its exact order is \begin{equation} 2N+3c(c+2) +\frac{r(3r+8c+5)-m(m+4c+1)}2. \label{p4:retained:eq:c17-relative-S-increment-order} \end{equation} Put \[ \chi_{c,m}=\frac{m(m+4c+1)}2,\qquad \psi_{c,r} =3c(c+2)+\frac{r(3r+8c+9)}2. \] Then \(s^{\rm in}_c\) defines a continuous one-way map \[ \mathscr S^{\rm in}_c: \prod_{m\geq0} J_mA^{\chi_{c,m}}\mathbb Q[[A]]Q_m^{\prime(c)} \longrightarrow \prod_{r\geq0} A^{\psi_{c,r}}\mathbb Q[[A]]Q_r^{\prime(c)}. \] Its \(r=0\) row is a continuous inside-disk cap functional. Moreover, \[ s^{\rm in}_{c;00} =(-1)^cA^{3c(c+2)}(1+O(A)), \] so the row becomes disk-primitive after the displayed monomial and unit normalization. \end{theorem} \begin{proof} The cutoff difference adds precisely the paths whose new intermediate index is \(N\). A unique lowest-order path controls the coefficient and the increment order rises with \(N\), giving \(A\)-adic convergence; the regulated input weights then make the row continuous. The resulting \(r=0\) map is an inside-disk cap on its declared domain, with no reverse transport asserted. A path contributing to \((\mathsf S_{c,N})_{rm}\), with intermediate indices \(i,j\), has exact order \[ \frac12\Bigl( 6c^2+12c+4i+4j+8cj+4j^2 -m-4cm-m^2+5r+8cr+3r^2 \Bigr). \] Subtracting the value at \(i=r,j=0\) gives \[ 2(i-r)+2j(2c+j+1). \] It is nonnegative and vanishes only at \(i=r,j=0\). Thus the lowest path is unique and cannot cancel. Evaluating it proves \eqref{p4:retained:eq:c17-relative-S-entry-order}. Increasing the cutoff from \(N-1\) to \(N\) adds exactly the paths with \(i=N\), proving \eqref{p4:retained:eq:c17-relative-S-cutoff-difference}. Among those paths \(j=0\) is uniquely minimal, and substitution gives \eqref{p4:retained:eq:c17-relative-S-increment-order}. Its order increases by two with each cutoff, proving coefficientwise convergence. Finally, \[ \ord_A\left(A^{\chi_{c,m}}s^{\rm in}_{c;rm}\right) =\psi_{c,r}. \] The \(m\)-th contribution from the regulated input domain has order at least \(\psi_{c,r}+2m\), since \(\ord_AJ_m=2m\). Hence each output sum converges continuously. The exact diagonal formula begins \[ b^{(c)}_{00}=(-1)^cA^{c(c+2)}(1+O(A)). \] At \(r=m=0\), the uniquely minimal path is \((b^{(c)}_{00})^3\), giving the displayed leading coefficient. \end{proof} The term \emph{one-way} is literal. The theorem proves a continuous map between the two displayed, generally different, weighted products. It does not assert surjectivity, a same-domain inverse, or a completed \(SL_2(\mathbb Z)\) action. Mirroring gives an \(A^{-1}\)-adic outside-sector candidate, but composing the two sectors requires a separate inside/outside endpoint-connection theorem. The resulting one-way row is the disk-side reading of a marked boundary state after the inside recapping operation. Its continuity records that the regulated state survives this reading in the declared chamber; it does not provide the reverse boundary transport or identify the disk cap with the cyclotomic cap. \subsubsection{The first-cyclotomic obstruction for every boundary color} \begin{theorem}[Raw cutoff word is never cyclotomic Cauchy] \label{p4:retained:thm:c17-relative-S-cyclotomic-obstruction} For every \(c\geq0\) and \(N\geq1\), \begin{equation} \left. (\mathsf S_{c,N}-\mathsf S_{c,N-1})_{00} \right|_{A=-1} =\frac{(c+\frac32)_N}{N!}\neq0. \label{p4:retained:eq:c17-relative-S-first-cyclotomic-value} \end{equation} Consequently, in these fixed coordinates, the raw cutoff sequence is not Cauchy even modulo \(1-A^2\), hence not in the cyclotomic or joint topology. Laurent monomial row and column gauges cannot remove this obstruction. \end{theorem} \begin{proof} The obstruction is detected before any completion argument: the first new intermediate color contributes a nonzero vacuum increment at $A=-1$. Because every multiple of $1-A^2$ vanishes there, the raw cutoff cannot be Cauchy even at the first cyclotomic level; Laurent monomial gauges only change the value by a sign and therefore cannot remove the defect. This isolates the failure of the raw word from the later question of constructing coherent counterterms or a unified row. For the vacuum increment, \[ (\mathsf S_{c,N}-\mathsf S_{c,N-1})_{00} =a^{(c)}_{0N}b^{(c)}_{N0}a^{(c)}_{00}. \] At \(A=-1\), \[ a^{(c)}_{00}\longrightarrow1,\qquad b^{(c)}_{N0}\longrightarrow(-2)^{-N}, \] and \[ \frac{\eta'_{c,N}}{\eta'_{c,0}} \longrightarrow 2^N\frac{(2c+2N+1)!!}{(2c+1)!!N!}. \] Since \[ a^{(c)}_{0N} =\frac{\eta'_{c,N}}{\eta'_{c,0}}b^{(c)}_{N0}, \] their product is \[ 2^{-N}\frac{(2c+2N+1)!!}{(2c+1)!!N!} =\frac{(c+\frac32)_N}{N!}. \] It is nonzero. Every multiple of \(1-A^2\) vanishes at \(A=-1\), while a Laurent monomial evaluates to \(\pm1\), proving both claims. \end{proof} Thus a joint construction needs either non-monomial cutoff counterterms with proved mapping-class coherence or an independently unified cyclotomic \(S\)-row combined with the disk row through the split joint completion. The theorem concerns every \(c,N\) for this raw cutoff family; it is not a no-go theorem for all bases or all non-monomial renormalizations. \subsubsection{Row, operator, and root base change} The following statements are distinct. \begin{enumerate} \item The \(r=0\) line of Theorem~\ref{p4:retained:thm:c17-relative-disk-S-arrow} is a continuous row on one declared disk domain. \item A generic mapping-class \(S\) must be an invertible arrow with a declared inverse domain and must satisfy the braid and central relations. The one-holed-torus relations include \[ t_\alpha t_\beta t_\alpha=t_\beta t_\alpha t_\beta,\qquad (t_\alpha t_\beta)^6=t_\delta. \] Those completed relations are not proved here. \item At an admissible odd-prime root, the finite matrix with \(N=d-c-1\) is separately invertible. Gilmer--Masbaum prove for their geometric lifts \[ (t t^\star t)^4 =q^{-6+2c(c+1)-p(p+1)/2}. \] Since \(q^p=1\), the final factor is rootwise one. The cyclotomic obstruction above shows that the finite root matrices are not evaluations of the raw inside-disk limit. \end{enumerate} \subsubsection{The fixed-marking arbitrary-genus consequence} Fix a lollipop marking \(G\), all spectator labels, and boundary half-colors \[ \mathbf c=(c_1,\ldots,c_g). \] Tensor the local arrows and put \[ \chi_{\mathbf c,\mathbf m} =\sum_{i=1}^g\chi_{c_i,m_i},\qquad \psi_{\mathbf c,\mathbf r} =\sum_{i=1}^g\psi_{c_i,r_i}. \] \begin{corollary}[Fixed-marking arbitrary-genus disk arrow] \label{p4:retained:cor:c17-fixed-marking-arbitrary-genus-S} For any proper height \(H_G\), the tensor matrix defines a continuous map \[ \widehat{\bigotimes}_{i=1}^g\mathscr S^{\rm in}_{c_i}: \prod_{\mathbf m} J_{H_G(\mathbf m)} A^{\chi_{\mathbf c,\mathbf m}} \mathbb Q[[A]]Q'_{\mathbf m} \longrightarrow \prod_{\mathbf r} A^{\psi_{\mathbf c,\mathbf r}} \mathbb Q[[A]]Q'_{\mathbf r}. \] \end{corollary} \begin{proof} With the marking and spectator sector fixed, tensoring the local inside-disk arrows makes the output order additive across handles. Properness of the height function turns the resulting lower-order estimate into a finite sum at each output coordinate, which is exactly the continuity statement. The argument stays within one marked decomposition and preserves the one-way inside-disk direction; it does not supply recoupling compatibility or a decomposition-independent handlebody theorem. Indeed, for fixed output \(\mathbf r\), an input term has order at least \[ \psi_{\mathbf c,\mathbf r}+2H_G(\mathbf m), \] and properness makes every bounded-height label set finite. \end{proof} This is an arbitrary-genus theorem for one fixed marked decomposition and fixed spectator sector. It is not a decomposition-independent handlebody theorem. For a recoupling arrow \(F:G\to G'\), the missing typed compatibility is \begin{equation} F^{\rm out}\mathscr S_G^{\rm in} =\mathscr S_{G'}^{\rm in}F^{\rm in}, \label{p4:retained:eq:c17-relative-S-F-compatibility} \end{equation} where \(F^{\rm in}\) and \(F^{\rm out}\) must be bicontinuous on the distinct\newline boundary-color-dependent input and output endpoint-valuation domains. The existing algebraic \(F\)-move theorems do not prove this: an \(F\)-move can mix the \(c\)-channels entering \(\chi_{c,m}\). A handlebody theorem also requires the \(S\)-specific stabilization law \[ \epsilon^{\rm in}_{g+1}(\operatorname{stab}v) =\epsilon^{\rm in}_{g}(v) \] on a proved lattice embedding. The existing contracted stabilization theorem applies to the previously constructed square-free contracted kernels, not this uncontracted infinite \(S\)-arrow. \paragraph{Primary source.} Gilmer--Masbaum, \emph{Integral TQFT for a one-holed torus} (arXiv:\allowbreak0908.2796), supplies the relative Newton bases, orthogonality and norm, the map\-ping-class presentation, Hopf adjointness and the identity \(a_{mn}^{(c)}=(\eta'_n/\eta'_m)b_{nm}^{(c)}\), and the root central relation. The exact generic norm factorization is in Theorem~\ref{p4:retained:thm:c11-norm}, and the exact positive-twist orders are in Section~\ref{p4:retained:ch:opposite-endpoints}. These formulas were also checked by exact symbolic computation. \paragraph{Precise status.} Proved: the all-\(c\) Newton rigging, exact cutoff difference, coefficientwise inside-disk convergence, continuous weighted domain, fixed-marking tensor extension, and all-\(c\) first-cyclotomic obstruction for the raw cutoff. Not proved: a same-domain inverse, the inside/outside endpoint connection, recoupling compatibility \eqref{p4:retained:eq:c17-relative-S-F-compatibility}, central relations in the completed disk category, stabilization of the new arrow, a cyclotomic relative row, root base change, or WRT/\(\widehat Z\) comparison. \subsection{What remains for a universal root-specialized scalar} Closed Habiro theory \cite{Habiro} supplies unified cyclotomic scalars for integral homology spheres and their WRT evaluations. It does not supply the boundary functor or the filling-dependent generic vacuum rows needed here; the homology-handlebody extension is explicitly a further construction. Even a completed closed cyclotomic cap would give only the cyclotomic coordinate, by Theorem~\ref{p4:retained:thm:c17-no-map}. For the closed-IHS part, Garoufalidis--L\^e already construct a cap on the full ordinary skein module \cite[Theorem~1.1, Theorem~2.4 and Section~3.1, arXiv v2]{GaroufalidisLe2023}. Their variable is \(q_{\mathrm{GL}}=Q=A^4\), so their fourth root is our Kauffman parameter \(A\). On a framed link with fundamental colors the cap is \[ [L]\longmapsto (-1)^{\sum_i(\operatorname{fr}(L_i)+1)}I_{(M,L)}(Q) \quad\text{in}\quad \mathscr H\otimes_{\mathbb Z[Q^{\pm1}]}\mathbb Z[A^{\pm1}], \] where \(\mathscr H\) is the ordinary Habiro ring. Arbitrary framed links span the source. Their Lemma~2.3 verifies the Kauffman relations term by term, and the convergent surgery sum preserves them; no restriction to algebraically split zero-framed retained links is needed. The even \(\mathbb Z[Q^{\pm1}]\)-sector lands in \(\mathscr H\), with empty-link value \(J_M\). This supplies a closed cap, not the filling-dependent boundary vacuum, mixed Hopf row, or joint completion required by the remaining theorem. The remaining theorem must construct, for every declared coefficient type, finite genus, handlebody double coset, and block label, a compatible vacuum and a mixed Hopf/$S$ row which: \begin{enumerate} \item is continuous on a declared primal lattice \(L_\phi\) and annihilates the correctly typed raw relation lattice \(L_\phi\cap\mathcal J_{\mathcal D_\phi}^{R}\); \item satisfies a proved complete presentation of the recoupling, bridge, twist, one-holed-torus Hopf/$S$, and central relations; \item obeys both handlebody covariances and the Maslov anomaly law; \item becomes integral under a stabilization gauge satisfying the exact normalization identities of Theorem~\ref{p4:retained:thm:c07-conditional-descent}; \item defines one joint disk--cyclotomic inverse-limit element; and \item has normalized WRT and intended $\widehat Z$ projections. \end{enumerate} The two genus-one filling-cap lines and their common polynomial approximants are now constructed by Theorem~\ref{p4:retained:thm:c17-rigged-two-cap}. In fixed Gilmer--Masbaum coordinates, Theorem~\ref{p4:retained:thm:c17-relative-disk-S-arrow} now constructs the all-boundary- color one-way disk arrow and its fixed-marking arbitrary-genus tensor extension. Theorem~\ref{p4:retained:thm:c17-relative-S-cyclotomic-obstruction} proves that its raw cutoffs cannot supply the cyclotomic coordinate, even at the first ideal. The smallest unresolved geometric case is therefore a cyclotomic relative row or non-monomial counterterm which is compatible with that disk arrow, together with the inside/outside connection, inverse, and central \(SL_2(\mathbb Z)\) relations. The typed recoupling equation \eqref{p4:retained:eq:c17-relative-S-F-compatibility} and an \(S\)-specific stabilization law must then make the fixed-marking tensor independent of the handlebody decomposition. The determinant obstruction shows why this cannot be a bare integral endomorphism of the ordinary color lattice.