\section{The disjoint-cube input and fixed-level quantum conventions}
\label{sec:standing-inputs}

The operator construction begins from the marked cube, its common homological
Lagrangian, and fixed Walker/TQFT conventions.  These data are transported,
not reproved.
We call the subset-zero datum the \emph{reference} datum; the word
``old'' is retained in formulas inherited from the source notation and has
the same meaning.

\subsection{Specified disjoint handleslide cubes with complete markings}

For \(n\geq1\), write \([n]=\{1,\ldots,n\}\).  Fix integers
\(g\geq1\), \(s\geq1\), and \(k\geq2s\).  Let
\[
  X=\Sigma_g^{\mathrm{out}},\qquad
  Y=\Sigma_{g+k}^{\mathrm{out}},\qquad d=g+k\geq3,
\]
and let \(W:X\to Y_{\mathrm{geom}}\) be a completed compression body with
\(k\) ordered product handles.  Fix an injective assignment
\(a\mapsto(i_a,j_a)\) from \([s]\) to ordered handle pairs such that all
\(2s\) active labels are distinct.  Assume that this family satisfies all
protected-sector hypotheses of the geometric cube theorem cited below.  In
particular, each of the corresponding \(s\) protected standard sectors carries
the complete local data and the prescribed positive product-zero-framed slide,
and their closures, including every protected product neighborhood and full
interface collar, are pairwise disjoint.  The one-holed-torus phase-support
collars of all \(k\) old marked rows are pairwise disjoint and disjoint from the
old-region marking.

Let \(\lambda_g\subset H_1(X;\mathbb Q)\) be the old disk-kernel
Lagrangian.  The old-region marking fixes a symplectic inclusion
\[
 \iota_X:H_1(X;\mathbb Q)\hookrightarrow H_1(Y;\mathbb Q),
 \qquad
 \lambda_g^Y:=\iota_X(\lambda_g).
\]
Let \(m_t,\ell_t\) be the meridian--longitude row of the \(t\)-th product
handle.  The complete old marking supplies pairwise-disjoint one-holed-torus
phase-support collars and the symplectic splitting
\begin{equation}
 H_1(Y;\mathbb Q)
 =\iota_X\bigl(H_1(X;\mathbb Q)\bigr)\oplus
  \bigoplus_{t=1}^k
   \operatorname{span}_{\mathbb Q}\{[m_t],[\ell_t]\}.
 \label{eq:standing-homology-splitting}
\end{equation}
Put
\begin{equation}
  \Lambda=\lambda_g^Y\oplus
  \bigoplus_{t=1}^{k}\mathbb Q[m_t]
  \subset H_1(Y;\mathbb Q).
  \label{eq:standing-lambda}
\end{equation}

We take \cite[Theorem~1.2]{MixedPhaseDisjointHandleslideCubes2026} as the following
independent geometric input.

\begin{inputtheorem}[Disjoint positive handleslide cube with complete markings]
\label{input:geometric-cube}
For every subset \(A\subseteq[s]\), there is a completed marked vertex
\(\mathfrak C_A\) and a parameterization \(p_A:\partial_+W\to Y\).  For
every \(a\notin A\), there is a supported orientation-preserving map
\(F_{a\mid A}:(W,\mathfrak C_A)\to(W,\mathfrak C_{A\cup\{a\}})\) and an
outgoing map
\begin{equation}
  h_{a\mid A}
  =p_{A\cup\{a\}}
   \bigl(F_{a\mid A}|_{\partial_+W}\bigr)p_A^{-1}.
  \label{eq:standing-edge-map}
\end{equation}
The maps have the identity germ on the full interface collar and fix the
incoming boundary, old region, other protected sectors, and unused handles
pointwise.  They
preserve \(\Lambda\) in both directions.  Hence
\begin{equation}
  H_{a\mid A}=C_Y(h_{a\mid A},0):(Y,\Lambda)\longrightarrow(Y,\Lambda)
  \label{eq:standing-walker-edge}
\end{equation}
is a well-typed weight-zero Walker cylinder.

For distinct \(a,b\notin A\), the supported maps form a literal
complete-marked square, and the corresponding cylinders satisfy
\begin{equation}
  H_{b\mid A\cup\{a\}}H_{a\mid A}
  =H_{a\mid A\cup\{b\}}H_{b\mid A}
  \label{eq:standing-source-square}
\end{equation}
with both composite weights zero.  The same package supplies the completed
endpoint phase cylinders, the independently typed forward and reverse
cobordisms, their edge covariances, finite order independence, and
telescoping, for arbitrary label-indexed integer weights.
\end{inputtheorem}

Its content is geometric: the square is established before any quantum
operator is formed.
In particular, the operator equalities below are one-way images of these
source equalities.

\subsection{Walker cylinders and Maslov weights}

The boundary surface is equipped with the rational Lagrangian \(\Lambda\).
For mapping classes \(f,g\) and integer weights \(m,n\), our convention is
\begin{equation}
 C_Y(g,n)C_Y(f,m)
 =C_Y\!\left(gf,n+m-
 \mu_Y(f_*\Lambda,\Lambda,(g^{-1})_*\Lambda)\right),
 \label{eq:standing-walker-product}
\end{equation}
with the right factor acting first; see
\cite[Equation~(2) and Section~9]{GilmerMasbaum2013} for this exact sign and
order convention, and \cite[Definitions~1.1--1.6]{Walker1991} for the
extended-cobordism framework.  Repeated Lagrangians give zero Maslov index, but
that vanishing is checked at each seam needed below rather than silently
assumed.

\subsection{Fixed-level TQFT, reduction, and universal construction}

We use the connected-surface Kauffman-bracket TQFT and handlebody
presentations of BHMV
\cite[Theorem~1.4 and Proposition~1.9]{BHMV1995}, in the
integral-weight/\linebreak[4]{}Lagrangian formulation of
\cite[Section~11, especially Remark~11.3]{GilmerMasbaum2013}.  The input is
the coefficient ring at the selected even root, finite state modules,
nondegenerate duality, surjective handlebody skein maps, and strict
functoriality on extended cobordisms.

We also use the accompanying positive rank,
weight-zero \(S^3\) normalization, and the selected anomaly branch.  Their
exact formulas are derived in Proposition~\ref{trfb1odmscoc:prop-root}; the
branch is not reconstructed from its square alone.

At a primitive \(4L\)-th root, reduction uses the local
\(f_{L-1}\)-relations, the finite admissible colored-spine basis, and the
nondegenerate complementary-handlebody bracket
\cite[Proposition~3.3(i) and Theorem~4.11]{BHMV1995}; trivalent evaluations
follow \cite{MasbaumVogel1994}.  The declared boundary Lagrangian remains
distinct from the actual disk kernel.

The universal-construction input is only the radical mechanism: a handlebody
skein is zero in the state module exactly when it pairs trivially with every
opposite handlebody state.  No odd-prime lattice or all-level completion is
imported.