\section{Root normalization, reduced fibres, and cube comparison}
\label{trfb1odmscoc:sec-root-reduced-crosswalk}

The first algebraic obstruction is finiteness: the geometric surface has a
natural skein module, but the operator theorem needs a finite fibre at one
fixed level.  This section constructs the normalized reduced source and one
comparison isomorphism from the reduced skein model to the fixed-level TQFT
state space.  That map is fixed once and reused at every vertex rather than
rebuilt along the cube.

We retain the quantifier order and notation fixed in
Section~\ref{sec:standing-inputs}.
In particular, the ordinary level \(K\) is fixed before a surface, a
handlebody, or a state space is constructed.  The coefficient ring and even-\(p\) fixed-level theory are kept together
with their handlebody skein surjections and duality.  The positive rank,
weight-zero \(S^3\) value, and anomaly branch fix how the complementary
pairing is normalized.  The geometric cube contributes only the reference
compression body.  On its skein module, the \(L\)-reduced quotient has a
finite admissible spine basis and a nondegenerate complementary bracket;
these are the ingredients that make the normalized comparison invertible.
The Walker boundary-Lagrangian and gluing-weight conventions remain in
force.  Surjectivity and TQFT duality identify the universal kernel with the
pairing radical.  The local reduced relations then make the map descend,
and nondegeneracy of the reduced complementary bracket proves injectivity.

\subsection{Root, ring, field, and the raw-field obstruction}

\begin{proposition}[Exact root and coefficient normalization]
\label{trfb1odmscoc:prop-root}
Fix \(K\geq1\), and put
\begin{equation}
 L:=K+2,\qquad p:=2L,\qquad
 \zeta_K:=\exp\!\left(\frac{\pi i}{2L}\right),\qquad
 A_K:=-\zeta_K .
\label{trfb1odmscoc:eq-root-data}
\end{equation}
Then \(A_K\) is a primitive \(4L\)-th root, the even-\(p\) palette is
\(\{0,\ldots,K\}\), and the anomaly data are
\begin{equation}
\begin{split}
 \mathcal D_K&=\frac{\sqrt{L/2}}{\sin(\pi/L)}>0,\\
 \Delta_K&=\mathcal D_K
   \exp\!\left(-\frac{3\pi iK}{4L}\right),\\
 \kappa_K&=\mathcal D_K\Delta_K^{-1}
   =\exp\!\left(\frac{3\pi iK}{4L}\right).
\end{split}
\label{trfb1odmscoc:eq-anomaly-data}
\end{equation}
Moreover,
\begin{equation}
 \kappa_K^2=A_K^{-6-p(p+1)/2},
\label{trfb1odmscoc:eq-root-square}
\end{equation}
where the positive-exponent formula in
\eqref{trfb1odmscoc:eq-anomaly-data}, rather than the square alone, fixes
the branch.  With
\begin{equation}
 F_K:=\mathbb Q(A_K),\qquad
 E_K:=\mathbb Q(A_K,\kappa_K),\qquad
 k_{p,K}:=\mathbb Z[1/p,A_K,\kappa_K]\subset E_K,
\label{trfb1odmscoc:eq-coefficient-rings}
\end{equation}
complex conjugation restricts to the involution
\(\iota_K(A_K)=A_K^{-1}\),
\(\iota_K(\kappa_K)=\kappa_K^{-1}\), and
\begin{equation}
 \boxed{\;
 Z_p(S^3,w=0)=\mathcal D_K^{-1}
 \in k_{p,K}\cap E_K^\times .
 \;}
\label{trfb1odmscoc:eq-r1f}
\end{equation}
For every connected extended surface \((Y,\Lambda)\), the central cylinder
\(W_Y=C_Y(\operatorname{id}_Y,1)\) satisfies
 \begin{equation}
 (E_K\otimes_{k_{p,K}}Z_p)(W_Y)
   =\kappa_K\operatorname{id}_{E_K\otimes_{k_{p,K}}V_p(Y,\Lambda)}.
\label{trfb1odmscoc:eq-central-weight}
\end{equation}
Finally, at \(K=1\) one has
\(\kappa_1\notin F_1\); hence \(F_K\) cannot uniformly replace \(E_K\).

After these data have been fixed, take \(g\geq1\), \(s\geq1\),
\(k\geq2s\), and \(d=g+k\geq3\), and import one datum supplied by
Input Theorem~\ref{input:geometric-cube}
with
\[
 X=\Sigma_g^{\mathrm{out}},\qquad
 Y=\Sigma_d^{\mathrm{out}},\qquad
 \Lambda=\lambda_g^Y\oplus
 \bigoplus_{t=1}^{k}\langle[m_t]\rangle_{\mathbb Q}.
\]
Here \(k\) counts handle sites and is unrelated to the coefficient ring
\(k_{p,K}\), while \(s\) counts slides and \([s]\) is the set of slide
indices.  Retain all source vertices \(\mathfrak C_A\), \(A\subseteq[s]\).
For every supplied vertex \(A\subseteq[s]\) and every \(a\notin A\),
retain, exactly as source data,
\[
 H_{a\mid A}=C_Y(h_{a\mid A},0),
\]
together with all endpoint phase cylinders, all completed forward and
reverse traces, all literal Walker squares, and the supplied common total
cylinder \(H_{[s]}\).  This section makes no assertion about their linear
images.  Fix also the low cap \(B_g\) with its fixed boundary
parameterization, regarded as the weight-zero extended cobordism
\(B_g^{(0)}:\varnothing\to(X,\lambda_g)\), with disk kernel
\(\lambda_g\), and distinguish
\begin{equation}
\begin{aligned}
 V^+_{p,K}&:=V_p(Y,\Lambda),&
 \mathcal V^+_{K,d}&:=E_K\otimes_{k_{p,K}}V^+_{p,K},\\
 V^-_{p,K}&:=V_p(X,\lambda_g),&
 \mathcal V^-_{K,g}&:=E_K\otimes_{k_{p,K}}V^-_{p,K}.
\end{aligned}
\label{trfb1odmscoc:eq-state-space-types}
\end{equation}
\end{proposition}

\begin{proof}
\emph{Anomaly branch.}
The normalization is BHMV's \(\eta\) and unknot formula in
\cite[Section~2, pp.~897--898, formula~(*) and the definition of \(\eta\)]{BHMV1995},
with \(p=2L\), palette \(0,\ldots,K\), and \(A=A_K\).
BHMV's anomaly parameter has cube \(\kappa_K\), the weight-one
character in \cite[Section~11]{GilmerMasbaum2013}; thus
\(\eta=\mathcal D_K^{-1}\) and
\(\langle U_{-1}(\Omega_p)\rangle=\Delta_K\).
With the displayed positive branch, the anomaly rank formula is the first
line of \eqref{trfb1odmscoc:eq-anomaly-data}.  Since \(L\geq3\),
\(\sin(\pi/L)>0\), so this is the positive rank.  The same calibration
gives
\[
 \Delta_K\mathcal D_K^{-1}
 =\exp\!\left(-\frac{3\pi i(L-2)}{4L}\right).
\]
As \(K=L-2\), this proves the second line and, after inversion, the third
line of \eqref{trfb1odmscoc:eq-anomaly-data}.  In particular it selects
\(\kappa_K\), not its negative.

\emph{Root-square identity.}
For the square convention, note first that
\(-1=\zeta_K^{2L}\), hence \(A_K=\zeta_K^{2L+1}\).  With \(p=2L\) and
\(K=L-2\), direct integer arithmetic gives
\begin{equation}
 (2L+1)\bigl(-6-L(2L+1)\bigr)-3(L-2)
 =-4L(L^2+L+4).
\label{trfb1odmscoc:eq-exponent-bridge}
\end{equation}
The two exponents are therefore congruent modulo the order \(4L\) of
\(\zeta_K\), and
\[
 A_K^{-6-p(p+1)/2}
 =\zeta_K^{\,3K}
 =\exp\!\left(\frac{3\pi iK}{2L}\right)
 =\kappa_K^2.
\]
This proves \eqref{trfb1odmscoc:eq-root-square} without weakening
\eqref{trfb1odmscoc:eq-exponent-bridge} to an unspecified congruence
representative.

\emph{Coefficient field and involution.}
The ring inclusion in \eqref{trfb1odmscoc:eq-coefficient-rings} is
literal: \(1/p\in\mathbb Q\subset E_K\), and both displayed generators
belong to \(E_K\).  Complex conjugation in \(\mathbb C\) inverts the two
roots of unity \(A_K\) and \(\kappa_K\), so it preserves \(E_K\) and has
the asserted values.

\emph{Vacuum normalization and central weight.}
The BHMV normalization \cite[Section~2, p.~898]{BHMV1995} identifies the invariant of the empty,
weight-zero \(S^3\) with
\(\mathcal D_K^{-1}\).  On the other hand the fixed-level TQFT is defined over
\(k_{p,K}\) and has \(V_p(\varnothing)=k_{p,K}\).  The same closed
cobordism therefore has its value in \(k_{p,K}\).  Equality with the
nonzero number \(\mathcal D_K^{-1}\) proves both the source-ring
membership and invertibility in \eqref{trfb1odmscoc:eq-r1f}; the
membership is not inferred merely from later scalar extension.
The weight convention of \cite[Section~11]{GilmerMasbaum2013} says that increasing a cobordism weight by one multiplies its
map by \(\kappa_K\).  Applying this to the identity cylinder and then
extending scalars proves \eqref{trfb1odmscoc:eq-central-weight}; integer
weights, including negative ones, act by the corresponding powers because
\(\kappa_K\in E_K^\times\).

\emph{Root order and reduced palette.}
Also,
\[
 \gcd(2L+1,4L)=\gcd(2L+1,2)=1,
\]
so \(A_K=\zeta_K^{2L+1}\) is primitive of order \(4L\).  The fixed-level TQFT
even-\(p=2K+4\) palette is consequently
\(\{0,\ldots,K\}\).  This proves the root-order assertion and fixes the
reduced convention used below.

\emph{The raw-field obstruction.}
It remains to check the raw-field obstruction.  For \(K=1\), \(L=3\),
\[
 A_1=-\zeta_{12}=\zeta_{12}^{\,7},\qquad
 F_1=\mathbb Q(\zeta_{12})=\mathbb Q(i,\sqrt3),\qquad
 \kappa_1=e^{\pi i/4}=\zeta_8.
\]
If \(\zeta_8\) belonged to \(F_1\), then
\(\sqrt2=\zeta_8+\zeta_8^{-1}\) would belong to the maximal real
subfield
\[
 F_1^+=\mathbb Q(\zeta_{12}+\zeta_{12}^{-1})
      =\mathbb Q(\sqrt3).
\]
But if \(\sqrt2=a+b\sqrt3\) with \(a,b\in\mathbb Q\), squaring forces
\(2ab=0\).  The alternatives would make either \(a^2=2\) or
\(b^2=2/3\), neither of which has a rational solution.  Thus
\(\sqrt2\notin\mathbb Q(\sqrt3)\), a contradiction.  Hence
\(\kappa_1\notin F_1\).

\emph{Geometric and module types.}
Finally, \(d=g+k\geq1+2=3\).  The surfaces, \(\Lambda\), and the
weight-zero source cylinders are the supplied geometric-cube data, while the cap
and its disk kernel are fixed before the high handlebody is formed.
Equation \eqref{trfb1odmscoc:eq-state-space-types} is then simply the
typed TQFT source-module/scalar-extension distinction.  This completes
the root, ring, field, anomaly, involution, and coefficient-field claims
stated above.
\end{proof}

\subsection{The old handlebody and its reduced fibre}

The old vertex is the anchor for all later TQFT transport.  Topologically, the
fixed low cap is attached to the reference compression body first; algebraically,
its disk kernel is identified and root-of-unity reduction is then imposed.
Only after the reduced fibre and its complementary pairing are available is
the common scalar extension used to define the comparison map.  This order is
why later vertices reuse one fixed comparison rather than acquiring separate
vertex-dependent maps.

\begin{proposition}[Old cap, disk kernel, reduction, and spine basis]
\label{trfb1odmscoc:prop-cap-red}
Glue the fixed low cap, through its fixed boundary parameterization, to
the common geometric-cube compression body in the reference chart
\(\mathfrak C_\varnothing\), and use the fixed old incoming and outgoing
parameterizations:
\begin{equation}
 H_{\varnothing,d}:=
 B_g\cup_X(W,\mathfrak C_\varnothing),\qquad
 \partial H_{\varnothing,d}\xrightarrow{\,p_\varnothing\,}Y.
\label{trfb1odmscoc:eq-old-handlebody}
\end{equation}
Then \(H_{\varnothing,d}\) is a genus-\(d\) handlebody, and, when assigned
weight zero,
\[
 H_{\varnothing,d}^{(0)}:
 \varnothing\longrightarrow(Y,\Lambda)
\]
has
\begin{equation}
 \ker\!\left(H_1(Y;\mathbb Q)\longrightarrow
 H_1(H_{\varnothing,d};\mathbb Q)\right)=\Lambda.
\label{trfb1odmscoc:eq-old-disk-kernel}
\end{equation}

There is a fixed parameterized complementary handlebody
\(H_{\varnothing,d}^{\mathrm c}\) such that
\(H_{\varnothing,d}\cup_YH_{\varnothing,d}^{\mathrm c}=S^3\).
It is used with weight zero as
\[
 (H_{\varnothing,d}^{\mathrm c})^{(0)}:
 \varnothing\longrightarrow(\overline Y,\Lambda).
\]
Here \(\Lambda\) is its \emph{declared boundary Lagrangian}; its actual
disk kernel is a different, complementary Lagrangian.

For one fixed banded trivalent spine
\(\Gamma_{\varnothing,d}\subset H_{\varnothing,d}\), let
\begin{equation}
 \mathcal S^{\mathrm{red}}_{K,d}
 :=\mathcal S_{A_K}(H_{\varnothing,d})/
 \langle f_{L-1}\text{-relations}\rangle .
\label{trfb1odmscoc:eq-reduced-quotient}
\end{equation}
Then
\begin{equation}
 \mathcal B_{K,d}
 =\{(\Gamma_{\varnothing,d})_c:
 c\text{ is \(L\)-admissible}\}
\label{trfb1odmscoc:eq-old-spine-basis}
\end{equation}
is a finite basis: every edge color is in \(\{0,\ldots,K\}\), every
vertex sum is even, the triangle inequalities hold, and the sum is at
most \(2K\).  The ordinary old fibre
\begin{equation}
 \mathcal M_{K,d}:=
 E_K\otimes_{F_K}\mathcal S^{\mathrm{red}}_{K,d}
\label{trfb1odmscoc:eq-reduced-fibre}
\end{equation}
is finite-dimensional, its displayed basis has nonzero double-handlebody
norms after scalar extension, and the reduced pairing with the
complementary handlebody is nondegenerate.

For \(U=B_g,H_{\varnothing,d},H_{\varnothing,d}^{\mathrm c}\), there is
the canonical base-change identification
\begin{equation}
 E_K\otimes_{k_{p,K}}\mathcal S_{k_{p,K}}(U)
 \cong
 E_K\otimes_{F_K}\mathcal S_{A_K}(U).
\label{trfb1odmscoc:eq-skein-base-change}
\end{equation}
\end{proposition}

\begin{proof}
Topologically, the reference compression body is obtained from
\(X\times[0,1]\) by the \(k\) fixed handle sites, whose co-cores have
old meridians \(m_1,\ldots,m_k\).  Gluing \(B_g\) to the incoming
boundary therefore attaches those \(k\) one-handles to the genus-\(g\)
handlebody.  The result is a genus-\(g+k=d\) handlebody, proving the first
assertion of \eqref{trfb1odmscoc:eq-old-handlebody}.

Every class in \(\lambda_g^Y\) is the old-region image of a class in
\(\lambda_g\) and dies in \(B_g\), while every \(m_t\) bounds
the corresponding co-core disk.  Thus \(\Lambda\) is contained in the
kernel in \eqref{trfb1odmscoc:eq-old-disk-kernel}.  The inclusion of the
boundary of a genus-\(d\) handlebody onto the handlebody is surjective on
rational first homology and has a \(d\)-dimensional kernel.  The direct
sum defining \(\Lambda\) has dimension \(g+k=d\).  The inclusion is
therefore an equality.  Assigning weight zero is part of the fixed
extended cobordism and introduces no scalar.

Choose the fixed standard complementary handlebody used by the reduced
pairing.  Walker's extended-cobordism formalism permits the boundary Lagrangian of an extended
three-manifold to be declared independently of its disk kernel, so its
boundary is declared to be \((\overline Y,\Lambda)\) and its weight is
set to zero \cite[Section~2]{GilmerMasbaum2013}.  Throughout, complementary
disk kernels are transported through
the identity-on-chains vector-space identification
\[
 \operatorname{or}_Y:H_1(\overline Y;\mathbb Q)
 \xrightarrow{\cong}H_1(Y;\mathbb Q)
\]
before they are compared with subspaces of \(H_1(Y;\mathbb Q)\) or entered
in a Maslov index; this identification reverses the intersection pairing.
To keep the two notions visibly distinct, write
\[
 \Lambda^{\mathrm c}:=
 \operatorname{or}_Y\!\left(
 \ker\!\left(H_1(\overline Y;\mathbb Q)\longrightarrow
 H_1(H_{\varnothing,d}^{\mathrm c};\mathbb Q)\right)\right)
 \subset H_1(Y;\mathbb Q).
\]
The Mayer--Vietoris sequence for
\(S^3=H_{\varnothing,d}\cup_YH_{\varnothing,d}^{\mathrm c}\) identifies
\(H_1(Y;\mathbb Q)\) with the direct sum of the first homologies of the
two handlebodies.  Hence
\(\Lambda\cap\Lambda^{\mathrm c}=0\), and both summands have dimension
\(d\).  In particular \(\Lambda^{\mathrm c}\neq\Lambda\); only the
declared boundary Lagrangian of the complement is \(\Lambda\).

Because \(A_K\) is primitive of order \(4L\), the reduction root hypothesis
holds with its source parameter \(r=L\).  Its negligible relation is
therefore exactly \(f_{L-1}\), and its allowed colors are
\(0,\ldots,L-2=K\).  Root-of-unity reduction gives the admissible colored-spine basis.
Its admissibility inequality is a strict vertex-sum inequality
\(<2L-2\); the sum is even, so this is equivalent to the bound
\(\leq2L-4=2K\) stated above.  A finite graph and a finite palette give
only finitely many such colorings.

The reduced-spine orthogonality formula expresses every diagonal norm as a
quotient of quantum dimensions and theta evaluations.  All quantum
integers occurring for an \(L\)-admissible coloring are nonzero at a
primitive \(4L\)-th root, and the theta formula is nonzero.  Thus every
norm is nonzero.  Its complementary-handlebody linking proposition makes
the bilinear bracket pairing nondegenerate.  Extending the corresponding
matrices from \(F_K\) to the field \(E_K\) preserves their nonzero
determinants, proving all assertions about
\(\mathcal M_{K,d}\).

Finally, a Kauffman-bracket skein module is presented by isotopy classes
of banded links and the local bracket relations, whose coefficients use
only \(A_K^{\pm1}\).  Extending either the \(k_{p,K}\)-presentation or the
\(F_K\)-presentation to their common field \(E_K\) therefore produces the
same \(E_K\)-module presentation.  The identity on link classes gives
\eqref{trfb1odmscoc:eq-skein-base-change}.  No spine or basis at any
other vertex has been introduced.
\end{proof}

\subsection{Quotient-first finiteness of the low state space}

The low state space is deliberately treated before coordinates.  The cap
first gives a surjection onto the TQFT fibre, while the reduced relations
produce a finite source; this establishes finiteness without selecting a low
spine basis or claiming injectivity.  The next factorization isolates the
universal-radical and complementary-pairing argument that will later justify
the fixed comparison map.

\begin{proposition}[Low-fibre finiteness before coordinates]
\label{trfb1odmscoc:prop-low-fin}
The weight-zero cap induces the typed surjection
\begin{equation}
\begin{aligned}
 q^-_{K,g}:\;
 E_K\otimes_{k_{p,K}}\mathcal S_{k_{p,K}}(B_g)
 &\twoheadrightarrow\mathcal V^-_{K,g},\\
 a\otimes x&\longmapsto
 a\otimes Z_p(B_g^{(0)},x)(1).
\end{aligned}
\label{trfb1odmscoc:eq-low-surjection}
\end{equation}
It kills every local \(f_{L-1}\)-relation and consequently factors as a
surjection
\begin{equation}
 E_K\otimes_{F_K}
 \mathcal S^{\mathrm{red}}_{A_K}(B_g)
 \twoheadrightarrow\mathcal V^-_{K,g},
 \qquad
 \boxed{\dim_{E_K}\mathcal V^-_{K,g}<\infty.}
\label{trfb1odmscoc:eq-low-fin}
\end{equation}
This conclusion chooses no low spine basis and asserts no injectivity.
\end{proposition}

\begin{proof}
 Handlebody surjectivity first presents the low state space as an image of
the skein module.  Complementary-handlebody testing makes the local
reduced relation vanish after gluing, so the map factors through the
reduced quotient; a finite admissible spanning family then proves
finite-dimensionality of the image without choosing a basis or claiming
injectivity.

The fixed-level TQFT handlebody presentation is surjective for every handlebody
whose boundary is the connected extended surface in question; it does
not require the declared boundary Lagrangian to equal the handlebody disk
kernel \cite[Remark~11.3]{GilmerMasbaum2013}.  Apply it to \(B_g^{(0)}\).
Extension of scalars is right exact,
so its scalar extension is precisely the surjection
\eqref{trfb1odmscoc:eq-low-surjection}.

Let \(n\) be a local relation containing \(f_{L-1}\).  Pair its TQFT
state with an arbitrary state of the oppositely oriented surface.
The latter state has a handlebody-skein representative by the same
TQFT surjectivity.  After gluing, the resulting closed skein still
contains the local \(f_{L-1}\)-relation.  The reduced vanishing trace says
that its evaluation is zero.  Since the TQFT duality pairing is
nondegenerate, the state of \(n\) is zero.  Thus
\eqref{trfb1odmscoc:eq-low-surjection} kills the entire negligible
submodule and factors through the reduced quotient.  Using
\eqref{trfb1odmscoc:eq-skein-base-change} gives the source displayed in
\eqref{trfb1odmscoc:eq-low-fin}.

The root-of-unity reduction argument supplies a finite spanning family indexed
by the \(L\)-admissible colorings of any finite handlebody spine.
Therefore the source of the factored map is finite-dimensional, and its
surjective image is finite-dimensional.  Only a spanning family was
used; no low basis, comparison map, or injectivity statement has been made.
\end{proof}

\subsection{The normalized universal pairing and one comparison map}

Set
\[
\begin{aligned}
 \mathscr U_{K,d}
 &:=E_K\otimes_{k_{p,K}}
   \mathcal S_{k_{p,K}}(H_{\varnothing,d}),\\
 \mathscr U^{\mathrm c}_{K,d}
 &:=E_K\otimes_{k_{p,K}}
   \mathcal S_{k_{p,K}}(H_{\varnothing,d}^{\mathrm c}),\\
 \mathcal V^{\mathrm c}_{K,d}
 &:=E_K\otimes_{k_{p,K}}V_p(\overline Y,\Lambda).
\end{aligned}
\]

\begin{proposition}[Exact pairing, factorization, and both kernels]
\label{trfb1odmscoc:prop-q-pair}
The two weight-zero handlebodies give maps
{\small
\begin{equation}
\begin{aligned}
 q^{\mathrm{univ}}_{K,d}:\mathscr U_{K,d}
 &\cong E_K\otimes_{F_K}
   \mathcal S_{A_K}(H_{\varnothing,d})
   \twoheadrightarrow\mathcal V^+_{K,d},&
 a\otimes x&\longmapsto
 a\otimes Z_p(H_{\varnothing,d}^{(0)},x)(1),\\
 q^{\mathrm c,\mathrm{univ}}_{K,d}:\mathscr U^{\mathrm c}_{K,d}
 &\cong E_K\otimes_{F_K}
   \mathcal S_{A_K}(H_{\varnothing,d}^{\mathrm c})
   \twoheadrightarrow\mathcal V^{\mathrm c}_{K,d},&
 b\otimes y&\longmapsto
 b\otimes
 Z_p((H_{\varnothing,d}^{\mathrm c})^{(0)},y)(1).
\end{aligned}
\label{trfb1odmscoc:eq-universal-maps}
\end{equation}
}
For \(a,b\in E_K\) and coefficient-ring skeins \(x,y\) in the displayed
handlebodies, the scalar-extended evaluation pairing is exactly
\begin{equation}
 \boxed{\;
 \operatorname{ev}_{Y,K}\!\left(
 q^{\mathrm{univ}}_{K,d}(a\otimes x),
 q^{\mathrm c,\mathrm{univ}}_{K,d}(b\otimes y)\right)
 =ab\,\mathcal D_K^{-1}
   \langle x\cup y\rangle_{A_K}.
 \;}
\label{trfb1odmscoc:eq-q-pair}
\end{equation}

If \(\mathscr N_{K,d}\subset\mathscr U_{K,d}\) denotes the
\(E_K\)-span of the local \(f_{L-1}\)-relations, then
\begin{equation}
 \mathscr N_{K,d}\subseteq
 \ker(q^{\mathrm{univ}}_{K,d})
 =\operatorname{Rad}_{\mathrm{left}}(\beta^{\mathrm{univ}}_{K,d}),
\label{trfb1odmscoc:eq-universal-radical}
\end{equation}
where
\(\beta^{\mathrm{univ}}_{K,d}(u,v)
=\operatorname{ev}_{Y,K}
(q^{\mathrm{univ}}_{K,d}(u),
 q^{\mathrm c,\mathrm{univ}}_{K,d}(v))\).
Consequently the first map in
\eqref{trfb1odmscoc:eq-universal-maps} descends to a surjection
\begin{equation}
 \overline q_{K,d}:\mathcal M_{K,d}
 \twoheadrightarrow\mathcal V^+_{K,d}.
\label{trfb1odmscoc:eq-q-factor}
\end{equation}
The two required kernel inclusions are, with every scalar extension
understood as displayed above,
\begin{equation}
\begin{split}
 \ker(q^{\mathrm{univ}}_{K,d})
 &\supseteq \mathscr N_{K,d},\\
 \ker(\overline q_{K,d})
 &\subseteq
 \operatorname{Rad}_{\mathrm{left}}
   \langle-,-\rangle_{\mathrm{red},E_K}=0 .
\end{split}
\label{trfb1odmscoc:eq-q-kernel-inclusions}
\end{equation}
\end{proposition}

\begin{proof}
The two kernel inclusions serve different purposes.  Vanishing of the local
reduced relations makes the universal map factor through the reduced
quotient; nondegeneracy of the reduced pairing proves that the descended
map is injective.  The exact gluing scalar connects the TQFT pairing to the
bracket used in both steps.

Surjectivity and the formulas in
\eqref{trfb1odmscoc:eq-universal-maps} are the fixed-level TQFT handlebody
presentation applied separately to \(H_{\varnothing,d}^{(0)}\) and to
\((H_{\varnothing,d}^{\mathrm c})^{(0)}\), followed by scalar extension.
This is why the source ring, the weight, the value on \(1\), and the
target scalar extension all appear explicitly.

We next compute the gluing normalization.  Let
\(\Lambda^{\mathrm c}\) be the actual disk kernel of the complementary
handlebody, as in the proof of Proposition
\ref{trfb1odmscoc:prop-cap-red}.  In the Walker gluing formula the three
Lagrangians are
\[
 \lambda_{H_{\varnothing,d}}(Y)=\Lambda,\qquad
 \lambda(Y)=\Lambda,\qquad
 \lambda_{H_{\varnothing,d}^{\mathrm c}}(\overline Y)
 =\Lambda^{\mathrm c}.
\]
Thus the glued weight is
\[
 0+0-\mu_Y(\Lambda,\Lambda,\Lambda^{\mathrm c})=0,
\]
because a Maslov index with two equal entries vanishes.  Notice that this
calculation uses the equality between the \emph{old} disk kernel and the
declared boundary Lagrangian; it does not identify the complementary disk
kernel with either one.

The glued closed manifold is consequently the weight-zero \(S^3\), with
skein \(x\cup y\).  The Kauffman-bracket skein module of \(S^3\) is
generated by the empty skein \cite[Theorem~2.3(a)]{Przytycki1999}, and the
class of \(x\cup y\) is
\(\langle x\cup y\rangle_{A_K}\) times that generator.  By
\eqref{trfb1odmscoc:eq-r1f}, its TQFT value is therefore
\[
 \mathcal D_K^{-1}\langle x\cup y\rangle_{A_K}.
\]
Multiplying the two scalar coefficients gives
\eqref{trfb1odmscoc:eq-q-pair}.  In particular the normalizing scalar is
the named element
\(\mathcal D_K^{-1}\in k_{p,K}\cap E_K^\times\), not an unspecified
unit.

For completeness, we now reconstruct the universal radical statement in
the present even-\(p=2K+4\) theory; compare
\cite[the universal construction following Proposition~1.1]{BHMV1995}.  If
\(u\in\ker(q^{\mathrm{univ}}_{K,d})\), then it pairs to zero with every
\(v\in\mathscr U^{\mathrm c}_{K,d}\), so it lies in the left radical of
\(\beta^{\mathrm{univ}}_{K,d}\).  Conversely, suppose \(u\) lies in
that radical.  The complementary map in
\eqref{trfb1odmscoc:eq-universal-maps} is surjective, so
\(q^{\mathrm{univ}}_{K,d}(u)\) evaluates to zero against every vector of
\(\mathcal V^{\mathrm c}_{K,d}\).  Nondegeneracy of the scalar-extended
TQFT duality pairing forces
\(q^{\mathrm{univ}}_{K,d}(u)=0\).  Hence
\[
 \ker(q^{\mathrm{univ}}_{K,d})
 =\operatorname{Rad}_{\mathrm{left}}(\beta^{\mathrm{univ}}_{K,d}).
\]
This is a purely universal-construction argument; no odd-prime lattice,
integrality, basis, or disconnected-surface theorem has entered it.

Now take \(n\in\mathscr N_{K,d}\).  For every complementary skein \(y\),
the closed bracket \(\langle n\cup y\rangle_{A_K}\) vanishes by the local
reduced trace identity for \(f_{L-1}\).  Equation
\eqref{trfb1odmscoc:eq-q-pair} therefore puts \(n\) in the universal
left radical, which is the kernel just proved.  This gives the first
inclusion in \eqref{trfb1odmscoc:eq-q-kernel-inclusions} and the
well-defined surjection \eqref{trfb1odmscoc:eq-q-factor}.
Indeed, \eqref{trfb1odmscoc:eq-skein-base-change} and right exactness of
scalar extension identify its source quotient explicitly as
\[
 \mathscr U_{K,d}/\mathscr N_{K,d}
 \cong E_K\otimes_{F_K}\mathcal S^{\mathrm{red}}_{K,d}
 =\mathcal M_{K,d}.
\]
The same local argument on the complementary side makes the bracket
descend to the two reduced quotients.

Finally let \(v\in\ker(\overline q_{K,d})\) and pair a representative
with every reduced complementary skein \(w\).  Equation
\eqref{trfb1odmscoc:eq-q-pair} and the nonzero scalar
\(\mathcal D_K^{-1}\) show that
\(\langle v,w\rangle_{\mathrm{red},E_K}\allowbreak=0\) for every \(w\).
The reduced complementary pairing is nondegenerate over \(F_K\), and
remains so after extension to \(E_K\).  Hence \(v=0\).  This proves the
second inclusion in \eqref{trfb1odmscoc:eq-q-kernel-inclusions}
independently of the first; no equality of dimensions has been used.
\end{proof}

\begin{proposition}[The comparison isomorphism at the old vertex]
\label{trfb1odmscoc:prop-q-iso}
The descended universal map is the isomorphism
\begin{equation}
 \boxed{\;
 Q_{K,d}:=\overline q_{K,d}:
 \mathcal M_{K,d}\xrightarrow{\ \sim\ }\mathcal V^+_{K,d}.
 \;}
\label{trfb1odmscoc:eq-q-iso}
\end{equation}
It is tied to the fixed low cap, old parameterization, coefficient
embedding, old weight-zero handlebody, old spine, and weight-zero
complementary testing handlebody.  For every source vertex and every
source edge, and for every later phase word, order word, and trace in
which the common state space occurs, the literal maps used are
\begin{equation}
 Q_{K,d}\quad\text{and}\quad Q_{K,d}^{-1};
\label{trfb1odmscoc:eq-global-q-choice}
\end{equation}
no vertex-indexed or route-indexed comparison map is introduced.
\end{proposition}

\begin{proof}
 The descended map is surjective by the quotient construction and injective
by the separately established complementary pairing.  Because its cap,
parameterization, coefficient embedding, spine, and testing handlebody are
all fixed at the old vertex, the same comparison map can be reused at every
source vertex and edge.  This is one fixed comparison, not a claim
of canonicity under changing those choices.

The map \(\overline q_{K,d}\) is surjective by
\eqref{trfb1odmscoc:eq-q-factor}.  Its kernel is zero by the second
inclusion in
\eqref{trfb1odmscoc:eq-q-kernel-inclusions}.  It is therefore the
isomorphism \eqref{trfb1odmscoc:eq-q-iso}, proving surjectivity and
injectivity by separate arguments.

Every object used to define this map was fixed at the old vertex and
depends on none of the vertex subsets.  The geometric cube theorem identifies every vertex
with the same abstract extended surface \((Y,\Lambda)\), so the codomain
is the same \(\mathcal V^+_{K,d}\) throughout.  Declaring
\eqref{trfb1odmscoc:eq-global-q-choice} to be the sole comparison map
therefore requires no compatibility or cancellation argument and creates
no new choice.  No canonicity under a change of cap, parameterization,
coefficient embedding, spine, or handlebody weight is asserted.
\end{proof}