\section{Attributed boundary inputs and the completion interface}
\label{p4:retained:ch:problem-conventions}

This paper starts from a marked Heegaard splitting
\(M_{\mathcal D}=H_\alpha\cup_\Sigma H_\beta\), a fixed trivalent spine in
\(H_\alpha\), and the Kauffman-bracket skein module over the coefficient ring
specified in each theorem.  Handle attachment, graph bases, and normalized
recoupling are used in the forms proved in
\cite{Przytycki,BHMV,MasbaumVogel,Costantino}; they are inputs rather than
new skein or TQFT constructions.

\subsection{The marked filling quotient}
\label{p4:retained:ch:arbitrary-genus-slide-quotient}

Attach the \(\beta\)-two-handles successively.  At each stage take every
slide difference through the new handle, pull the resulting kernel back to
\(\Sk_A(H_\alpha;R)\), and denote the sum of these pulled-back kernels by
\(\mathcal J_{\mathcal D}^{R}\).  Later relations are therefore imposed in
the already-filled manifold; a hand-picked list of slides in the initial
handlebody is not substituted for this successive kernel.

The one-handle quotient used at each stage is
\cite[Proposition~2.2(5)]{Przytycki}.

\begin{theorem}[Marked filling quotient]
\label{p4:retained:thm:c08-all-genus-quotient}
For every \(g\geq0\), inclusion induces a pointed isomorphism
\[
 \bigl(\Sk_A(H_\alpha;R)/\mathcal J_{\mathcal D}^{R},[\varnothing]\bigr)
 \cong
 \bigl(\Sk_A(M_{\mathcal D};R),[\varnothing]\bigr).
\]
The result is independent of the order of the disjoint
\(\beta\)-attachments.
\end{theorem}

\begin{proof}
Przytycki's Proposition~2.2(5) identifies the skein module after one
two-handle attachment with the quotient by all slide differences through
that handle.  Apply this result successively to the ordered disjoint
\(\beta\)-curves.  The kernel of the composite map is the sum of the stage
kernels pulled back to the initial handlebody module, which is exactly
\(\mathcal J_{\mathcal D}^{R}\).  After all two-handles are attached, the
remaining boundary spheres are capped by three-handles; these attachments
induce skein-module isomorphisms by
\cite[Proposition~2.2(2)(i)]{Przytycki}.  The composite therefore gives the
displayed pointed isomorphism.

Changing the attachment order gives the same filled manifold.  For every
order, the composite is the inclusion-induced map to its skein module, so
the resulting quotient isomorphism is independent of that order.
\end{proof}

\subsection{Cyclotomic and disk completions}

Put
\[
 (q;q)_N=\prod_{k=1}^{N}(1-q^k),\qquad
 \widehat{\Z[q]}_{\mathrm{cyc}}
 =\varprojlim_N\Z[q]/((q;q)_N),\qquad
 \widehat{\Z[q]}_{\mathrm{disk}}=\Z[[q]].
\]
The first ring is Habiro's cyclotomic completion \cite{Habiro}; the second
uses the \(q\)-adic topology.  They are retained as distinct coordinates.

\begin{theorem}[No continuous polynomial-identity extension]
\label{p4:retained:thm:c07-no-continuous-map}
There is no continuous additive map
\[
 \Psi:\widehat{\Z[q]}_{\mathrm{cyc}}\longrightarrow\Z[[q]]
\]
whose restriction to \(\Z[q]\) is the ordinary polynomial inclusion.  In
particular, no continuous ring homomorphism with this property exists.
\end{theorem}

\begin{proof}
The sequence \(P_N=(q;q)_N\) tends to zero cyclotomically.  If \(\Psi\)
were continuous and fixed polynomials, then \(\Psi(P_N)=P_N\) would tend to
zero \(q\)-adically, but every \(P_N\) has constant coefficient one.
\end{proof}

For a module \(M\), the finite ideals used below are
\(\mathfrak j_N=(A^{2N}P_N)\).  Their Chinese-remainder splitting records two
residues of one finite representative; it does not select either completed
coordinate from the other.

\subsection{Conditional closed descent}

The later cap discussion uses the following formal implication.  An
\emph{enhanced Heegaard datum} consists, in every genus, of chambered right
and left modules and vacua \(\Omega_g^\pm\), a continuous pairing, an exact
representation of the declared central extension, coherent lifts of both
handlebody subgroups, an anomaly function \(\omega_g\), and stabilization
data \(u_g,s_g,\nu_g\) satisfying
\[
 A_{g+1}(\widetilde{\operatorname{st}}_g\widetilde\phi)
 =u_g\kappa^{s_g}A_g(\widetilde\phi),\qquad
 \omega_{g+1}=\omega_g+s_g,\qquad \nu_{g+1}u_g=\nu_g,
\]
where
\(A_g(\widetilde\phi)=
\langle\Omega_g^-,\rho_g(\widetilde\phi)\Omega_g^+\rangle\).
All products and pairings must exist in the stated chambers.

\begin{theorem}[Conditional enhanced Heegaard descent]
\label{p4:retained:thm:c07-conditional-descent}
If an enhanced Heegaard datum exists for a class of closed decorated
three-manifolds, then
\[
 \mathfrak Z(M_\phi)=
 \nu_g\kappa^{-\omega_g(\widetilde\phi)}
 \langle\Omega_g^-,\rho_g(\widetilde\phi)\Omega_g^+\rangle
\]
is independent of the central lift, left and right handlebody mapping
classes, and Heegaard stabilization.  It therefore depends only on the
oriented decorated three-manifold, and every declared cyclotomic or chamber
projection is separately topological.
\end{theorem}

\begin{proof}
Replacing \(\widetilde\phi\) by \(z^n\widetilde\phi\) multiplies the
matrix element by \(\kappa^n\) and increases \(\omega_g\) by \(n\), so
the two factors cancel.  Left and right handlebody changes disappear by the
two covariance identities, while \(\omega_g\) is constant on the same
double coset.  Under stabilization the scalar factor is
\[
 \nu_{g+1}\kappa^{-\omega_g-s_g}u_g\kappa^{s_g}
   =\nu_g\kappa^{-\omega_g}.
\]
Stable equivalence of Heegaard splittings then proves topological
invariance.  Continuity makes every declared projection commute with the
matrix element.
\end{proof}

This theorem is an implication, not an existence result.

\subsection{Recoupling transport}
\label{p4:retained:ch:recoupling-groupoid}

Fix a base marking \(G_0\), a base chamber \(\alpha\), and its primal and
dual lattices.  For a verified recoupling degree \(\gamma\), put
\[
 L_{\gamma,\alpha}=\rho(\gamma)L_{0,\alpha},\qquad
 L^\vee_{\gamma,\alpha}=(\rho(\gamma)^{-1})^*L^\vee_{0,\alpha}.
\]

\begin{theorem}[Chamber local system]
\label{p4:retained:thm:c14-local-system}
For every finite genus, each verified arrow \(a\) gives exact
\(\Lambda\)-linear identifications
\[
 \rho(a):L_{\gamma,\alpha}\xrightarrow{\sim}L_{a\gamma,\alpha},
 \qquad
 (\rho(a)^{-1})^*:L^\vee_{\gamma,\alpha}
 \xrightarrow{\sim}L^\vee_{a\gamma,\alpha}.
\]
They preserve every scalar submodule \(J_N\), extend bicontinuously to the
completions, and pure recoupling paths with the same endpoints give identical
transported lattices and test functionals.
\end{theorem}

\begin{proof}
Transport the order, meet, and join from the base chamber.  The module
identity is
\(\rho(a)\rho(\gamma)L_0=\rho(a\gamma)L_0\), and the dual identity is its
contragredient.  Scalar ideals commute with every linear transport, giving
the finite-stage preservation and bicontinuous extensions.  Finally,
normalized recoupling coherence \cite{MasbaumVogel,Costantino} identifies
pure recoupling paths with the same endpoints, so the transported lattices
and test functionals are path independent.
\end{proof}

No native lattice at a new marking is identified without the separately
proved bridge below.

\subsection{Homological data of a Heegaard splitting}
\label{p4:retained:ch:heegaard-homology}
\label{p4:retained:chap:heegaard-homology}

Choose a symplectic basis
\begin{equation}
  (a_1,\ldots,a_g,b_1,\ldots,b_g)
\end{equation}
in which the \(a_i\) bound disks in the first handlebody, and write
\begin{equation}
  \phi_*=\begin{pmatrix}A&B\\ C&D\end{pmatrix}
  \in\operatorname{Sp}(2g,\mathbb Z).
  \label{p4:retained:eq:symplectic-blocks}
\end{equation}
Let \(L_A\subset H_1(\Sigma_g;\mathbb Z)\) be the meridian Lagrangian of
the first handlebody.  Mayer--Vietoris gives
\begin{equation}
  H_1(M_\phi;\mathbb Z)
  \cong H_1(\Sigma_g;\mathbb Z)/(L_A+\phi_*L_A).
  \label{p4:retained:eq:MV-lagrangians}
\end{equation}
Modulo \(L_A\), the relations contributed by \(\phi_*L_A\) are the columns
of \(C\).  Thus the direct quotient is the column model
\(G_{\rm col}=\mathbb Z^g/C\mathbb Z^g\); the row model below is its
canonical coordinate counterpart.

\begin{theorem}[Homological data of the Heegaard splitting]
\label{p4:retained:thm:heegaard-homological-shadow}
For the block convention of \eqref{p4:retained:eq:symplectic-blocks},
\begin{equation}
  H_1(M_\phi;\mathbb Z)
  \cong G_{\rm col}:=\operatorname{coker}(C)
  \cong G_{\rm row}:=\operatorname{coker}(C^T).
  \label{p4:retained:eq:H1-coker-C}
\end{equation}
The canonical isomorphism \(G_{\rm row}\to G_{\rm col}\) is
\([u]\mapsto[Du]\), with inverse \([x]\mapsto[A^Tx]\).
If \(\det C\ne0\), then \(M_\phi\) is a rational homology sphere,
\begin{equation}
  |H_1(M_\phi;\mathbb Z)|=|\det C|,
\end{equation}
and the torsion linking form is represented in the row model by
\begin{equation}
  Q_\phi=C^{-1}D\pmod{\mathbb Z}.
  \label{p4:retained:eq:heegaard-linking-form}
\end{equation}
In particular, \(Q_\phi\) is symmetric.
\end{theorem}

\begin{proof}
The first isomorphism is the quotient computation following
\eqref{p4:retained:eq:MV-lagrangians}.  The map \([u]\mapsto[Du]\) from
the row model is well defined because \(DC^T=CD^T\); \([x]\mapsto[A^Tx]\)
is well defined because \(A^TC=C^TA\).  The identities
\(A^TD-C^TB=I\) and \(DA^T-CB^T=I\) show that the two induced maps are
inverse.  Nonsingularity of \(C\) is equivalent to finiteness, and the group
order is \(|\det C|\).

In the direct \(b\)-column coordinates, the Heegaard linking calculation
gives
\[
 \operatorname{lk}_{\rm col}([x],[y])
 =x^TAC^{-1}y\pmod{\mathbb Z}.
\]
Indeed, if \(Cu=nx\), the corresponding combination of meridian disks has
boundary \(nx\) after its \(Au\) part is capped in the first handlebody;
dividing its intersection with \(y\) by \(n\) gives the displayed matrix in
the fixed orientation.  It descends because \(AC^{-1}\) is symmetric and
\((AC^{-1})C=A\) is integral.  Pulling the form back by \(\overline D\)
gives
\[
 u^TD^TAC^{-1}Dv.
\]
Since \(D^TA-B^TC=I\),
\[
 D^TAC^{-1}D=C^{-1}D+B^TD
 \equiv C^{-1}D\pmod{\operatorname{Mat}_g(\mathbb Z)}.
\]
Thus \eqref{p4:retained:eq:heegaard-linking-form} is the row-coordinate
matrix.  It descends directly because \(C(C^{-1}D)=D\), and symmetry follows
from \(CD^T=DC^T\).
\end{proof}

This theorem records homological data only; it does not identify the
trivalent-spine operator or any TQFT amplitude.