\section{Attributed cap and sewing inputs}

The remaining closed comparisons use three established constructions rather
than introducing a second boundary theory.  First, stated-skein excision and
Morita sewing give the generic Heegaard balanced tensor product
\cite{CostantinoLe}.  Second, on the integral-homology-sphere locus, the
ordinary skein module admits the Habiro-type closed cap of
\cite[Theorem~1.1 and Section~3.1, arXiv v2]{GaroufalidisLe2023}.
Third, the universal-construction quotient by the
gluing-pairing radical is the standard BHMV mechanism \cite{BHMV}.

For the fixed variance, marking, and coefficient conventions of this paper,
these inputs yield a generic boundary bimodule, its declared closed cap, and
the cap-induced radical quotient.  The only comparison asserted here is
after that cap or sewing map has been applied.  In particular:
\begin{enumerate}
\item the stated-skein/Morita construction is not extended to the integral
root fibers of the rootwise construction;
\item the Habiro cap is used only on the stated integral-homology-sphere
domain and gives no evaluation map on the whole generic localization; and
\item the cap-induced radical quotient is not identified with the
perturbative or operational boundary radicals.
\end{enumerate}

The cited primary sources contain the imported excision, cap, and
universal-construction theorems.  The nontrivial perturbative coefficient
comparison proved in the preceding section remains part of this paper.

\subsection{Closed-cap comparisons for Heegaard scalars}
\label{p4b:sec:public-closed-cap-interfaces}

The following two propositions give the closed-cap comparisons used by the
companion Heegaard-certificate construction
\cite{ClosedHeegaardDescentRadialZhat2026}.  Their proofs are included here.
Let \(A=Q^{1/4}\), let
\[
 \mathscr H_A
 =\mathscr H\otimes_{\mathbb Z[Q^{\pm1}]}\mathbb Z[A^{\pm1}],
\]
and, for an oriented integer homology sphere \(M\), let
\[
 \varphi_M:\mathcal S_{\mathbb Z[A^{\pm1}]}(M)\longrightarrow\mathscr H_A
\]
be the ordinary-skein cap of Garoufalidis--L\^e
\cite[Theorem~1.1, Theorem~2.4 and Section~3.1, arXiv v2]{GaroufalidisLe2023}.
Their variable is \(q_{\mathrm{GL}}=Q=A^4\); their
\(q_{\mathrm{GL}}^{1/4}\) is our Kauffman parameter \(A\).
For the stated-skein comparison, the Costantino--L\^e parameter is
\(A\), the Korinman--Murakami parameter is \(A^2\), and the stated
ground ring and carrier field are respectively
\(\mathbb Z[A^{\pm1/2}]\) and \(\mathbb Q(A^{1/2})\).
On a framed unoriented link \(L=L_1\cup\cdots\cup L_r\), the cap is
\[
 \varphi_M([L])=
 (-1)^{\sum_i(\operatorname{fr}(L_i)+1)}I_{(M,L)}(Q),
\]
where every component has the fundamental color and
\(\operatorname{fr}(L_i)=\operatorname{lk}(L_i,L_i)\).
This fixes the vacuum sign and the loop value
\(\varphi_{S^3}(\bigcirc)=-(A^2+A^{-2})\).

All framed links generate the ordinary skein module.  Theorem~2.4 of the
cited source constructs the pair invariant on this full IHS domain; it is
not restricted to algebraically split zero-framed retained links.
Lemma~2.3 and Section~3.1 prove that each signed colored-Jones functional
in the convergent surgery sum satisfies the Kauffman crossing relation
with coefficients \(A,A^{-1}\) and the loop relation above.  Linear
extension and passage to the Habiro-convergent sum therefore give the
displayed module map through both skein relations.

Write
\(\mathcal S^{\mathrm{ev}}(M)\) for the
\(\mathbb Z[Q^{\pm1}]\)-span of root-lattice-colored links, equivalently
two-parallels.  This pure sector is a \(\mathbb Z[Q^{\pm1}]\)-module;
multiplication by \(A\) is not part of its definition.

\begin{proposition}[Pure Habiro sector and vacuum]
\label{p4:prop:pure-habiro-vacuum}
The even restriction lands in the ordinary Habiro ring
\[
 \varphi_M^{\mathrm{ev}}:
 \mathcal S^{\mathrm{ev}}(M)\longrightarrow\mathscr H.
\]
The empty skein belongs to this sector and
\begin{equation}
 \varphi_M(\varnothing)=J_M.
 \label{p4:eq:cap-vacuum}
\end{equation}
At every root \(\xi\), \(J_M(\xi)\) is Habiro's normalized \(SU(2)\) WRT
invariant.  When the chosen fourth root \(A=\xi^{1/4}\) is itself primitive
of odd order, this agrees with the normalized \(SO(3)\) specialization of
the ordinary-skein cap.  With \(Q=e^h\),
\[
 \operatorname{Exp}_1(\iota_1(J_M))
 =\widehat W_{\mathfrak{sl}_2}^{\mathrm{tr}}(\Omega(M)).
\]
\end{proposition}

\begin{proof}
For even highest weights the exponent \(g/4\) in Theorem~2.4 of
\cite{GaroufalidisLe2023} is integral.  Its representation-ring extension
therefore lands in the ordinary Habiro ring on the even sector.  For the empty link the sign
is one and the pair invariant is \(J_M\).  Habiro's all-root theorem
\cite{Habiro} gives the root specializations, while
Theorem~\ref{p4:thm:lmo-habiro-closed-bridge} gives the Taylor/LMO
comparison.  These are precisely the asserted restrictions and scalar
identities.
\end{proof}

Put
\[
 \mathcal K=\mathscr H_A\otimes_{\mathbb Z[A^{\pm1}]}\mathbb Q(A^{1/2}).
\]
This is a commutative coefficient extension, not a declared field.
The generic closed cap is obtained by scalar extension, as follows.
The endpoint-free ordinary-to-punctured-stated map becomes an isomorphism
over \(\mathbb Q(A^{1/2})\), using the spherical comparison of
\cite{KorinmanMurakami} and the filled/endpoint-free sphere lemma of
\cite{CostantinoLe}.  Its integral kernel is torsion.  The Habiro target
\(\mathscr H_A\) is torsion-free over \(\mathbb Z[A^{\pm1}]\):
its Taylor embedding at \(Q=1\), together with the free fourth-root
coefficient extension, detects multiplication by a nonzero Laurent
polynomial.  Thus \(\varphi_M\) annihilates that kernel and descends to
the image lattice in the generic carrier.  This lattice spans the carrier,
so scalar extension gives its \(\mathcal K\)-valued cap.  This argument
uses neither an integral spherical isomorphism nor a root evaluation on
all of \(\mathcal K\).  Root values in the first proposition are taken
on the integral Habiro sector before this generic extension.

On the raw empty-skein geometric domain, the operational coefficient is the
Habiro cap of the oriented closed composite, while the generic GNS
coefficient is the scalar obtained by applying the same cap after
Costantino--L\^e sewing.  The latter category is the same-oriented opposite
of the CHM convention used by the operational construction.

\begin{proposition}[Variance-reversed operational agreement]
\label{p4:prop:gns-operational-agreement}
On the raw empty-skein geometric domain, after the coefficient inclusion
\(\mathscr H\hookrightarrow\mathcal K\), the operational pairing and the GNS
pairing are the same \(J_M\).  The identification reverses state/\allowbreak cotest
variance by the same-oriented opposite equivalence.  It uses neither
orientation reversal of the three-manifold nor \(Q\mapsto Q^{-1}\).
\end{proposition}

\begin{proof}
The comparison is made before either radical quotient.  A CHM state
\(0\to S\) corresponds to a Costantino--L\^e cotest \(S\to0\),
and a CHM cotest to a state, in the same-oriented opposite category.
Both constructions
glue the same oriented geometric cup, middle cobordism, and cotest, and both
then apply the closed cap to that composite.  On elementary balanced
tensors this is the same sewn tangle followed by the same linear cap;
the balancing relations give the same scalar contraction.  Empty skeins sew to the empty
skein, whose value is \(J_M\) by
Proposition~\ref{p4:prop:pure-habiro-vacuum}.  Passing to the same-oriented
opposite category interchanges the state and cotest roles but does not
reverse the glued manifold and does not change \(Q\).  After the displayed
coefficient inclusion, the two scalar pairings are therefore identical.
\end{proof}

The second proposition is deliberately restricted to the raw geometric
domain.  It does not identify the operational and generic GNS radicals.