\section{Introduction} How does a marked handleslide relation become a literal finite-level operator relation? One normalized skein-to-TQFT comparison provides a common coordinate model and avoids carrying vertex-dependent conjugations. A projective mapping-class action would retain the relation only up to an anomaly scalar. We construct one normalized comparison isomorphism, reuse it at every vertex, and retain the central factors by working in Walker's extended cobordism category. The geometric cube and pentagon are supplied by Papers~1 and~2. Their literal Walker equalities become literal \(E_K\)-linear operator equalities under this fixed comparison. The contribution here is its construction and exact normalization, coefficient and anomaly control, and the typed groupoid comparing auxiliary choices. The chosen isomorphism, called a comparison map below, relates two specified models; it is not a canonical identification or a choice that may vary from vertex to vertex. \subsection{From marked topology to the finite TQFT fibre} The source relation lives in Walker's extended cobordism category. Its marked representatives determine the geometric composites, its Lagrangian data determine the Maslov corrections, and its integer weights retain the central anomaly. At a fixed root, the handlebody skein module is reduced by the negligible-color relations and compared with the finite TQFT state space through the universal construction. At fixed \(K\), the universal handlebody map factors through the \(L\)-reduced skein quotient. The exact weight-zero \(S^3\) pairing, normalized by \(\mathcal D_K^{-1}\), proves both injectivity and surjectivity, giving the isomorphism \(Q_{K,d}\). Strict functoriality then transports the already proved geometric relation without changing factor order or discarding its weight. Cyclotomic arithmetic enters only to fix the root, coefficient field, and anomaly branch. Completion theory and disk \(q\)-series belong to separate all-level questions and are not proof dependencies here. \begin{figure}[htbp] \centering \begin{tikzpicture}[ obj/.style={draw,rounded corners=3pt,inner sep=4pt}, arr/.style={-{Latex[length=2mm]},thick}, every node/.style={font=\small} ] \node[obj] (M1) at (0,1.2) {\(\mathcal M_{K,d}\)}; \node[obj] (M2) at (4.2,1.2) {\(\mathcal M_{K,d}\)}; \node[obj] (V1) at (0,0) {\(\mathcal V^+_{K,d}\)}; \node[obj] (V2) at (4.2,0) {\(\mathcal V^+_{K,d}\)}; \draw[arr] (M1) -- node[above] {\(\rho^Q_{K,d}(G)\)} (M2); \draw[arr] (V1) -- node[above] {\(Z_K(G)\)} (V2); \draw[arr] (M1) -- node[left] {\(Q_{K,d}\)} (V1); \draw[arr] (M2) -- node[right] {\(Q_{K,d}\)} (V2); \node[obj,align=center,font=\scriptsize] (source) at (7.3,1.65) {\(H_{b\mid A\cup\{a\}}H_{a\mid A}\)\\ \(=H_{a\mid A\cup\{b\}}H_{b\mid A}\)}; \node[obj,align=center,font=\scriptsize] (target) at (7.3,-.65) {\(T^K_{b\mid A\cup\{a\}}T^K_{a\mid A}\)\\ \(=T^K_{a\mid A\cup\{b\}}T^K_{b\mid A}\)}; \draw[arr] (source) -- node[right=3pt,align=left,font=\scriptsize] {apply \(Z_K\),\\then conjugate\\by \(Q_{K,d}\)} (target); \end{tikzpicture} \caption{The fixed comparison square and the transported coherence relation. The source equality is a weight-zero Walker equality proved geometrically. The same \(Q_{K,d}\) is used on every edge; the diagram has no reverse implication and makes no comparison between levels.} \label{fig:geometric-to-tqft-transport} \end{figure} \begin{headlinetheorem}[Fixed-level transport of the disjoint handleslide cube] \label{thm:intro-cube-transport} Fix \(K\geq 1\) and a supplied marked disjoint handleslide cube from Paper~1. There is one normalized comparison isomorphism \(Q_{K,d}\), reused at every vertex, such that each geometric edge is sent to an \(E_K\)-linear automorphism and every weight-zero geometric face is sent to a literal operator square. The same transport preserves the finite route equalities and the declared forward/reverse phase and weight telescoping identities. It is a fixed-level image of the geometric cube and makes no comparison between different levels. \end{headlinetheorem} The complete normalization and edge statements are in Theorem~\ref{trfb1odmscoc:thm-T0}. Its geometric input is \cite[Theorem~1.2]{MixedPhaseDisjointHandleslideCubes2026}. The pentagon reuses three handles, so its overlapping directions cannot be treated as a disjoint cube. Its geometric source is \cite[Theorem~4.1, Proposition~12.3, Theorems~13.5 and~14.1, and Section~15]{CompleteMarkedPentagons2026}. Every selected vertex-zero datum admits a rational disk-kernel cap, a weight-zero \(S^3\) complement, a reduced spine model, and a single comparison isomorphism. Two such models compare through their common TQFT fibre and intertwine every Walker endocylinder. Applying this comparison core and the strict-functor theorem directly to the geometric Walker pentagon gives the second result. \begin{headlinetheorem}[Fixed-level transport of the marked handleslide pentagon] \label{thm:intro-pentagon-transport} Fix \(K\geq 1\) and a selected marked three-handle pentagon from Paper~2. A compatible comparison core transports its weight-zero Walker relation to the literal \(E_K\)-linear operator equality \[ \Tcore{B\star}{\mathfrak q}\Tcore{0B}{\mathfrak q} =\Tcore{AC\star}{\mathfrak q}\Tcore{AAC}{\mathfrak q} \Tcore{0A}{\mathfrak q}, \] with no projective scalar. Comparison cores for the same TQFT fibre give intertwined operator systems and satisfy the corresponding comparison groupoid law; no canonical coordinate choice is asserted. \end{headlinetheorem} Theorem~\ref{thm:operator-pentagon} gives the operator equality, and Theorem~\ref{thm:crosswalk-choice-comparison} gives the change of comparison data. These complement the finite-order relations for independent supports without inferring overlapping coherence from disjoint squares. \subsection{Established inputs and scope} Przytycki's skein-module framework provides the algebraic starting point \cite[Definition~2.1]{Przytycki1999}, while BHMV supplies the Kauffman-bracket TQFT, universal construction, and handlebody presentations \cite[Theorem~1.4 and Proposition~1.9]{BHMV1995}. At the chosen root, admissibly colored banded trivalent graphs give finite state-space coordinates in the BHMV theory \cite[Theorem~4.11]{BHMV1995}; the evaluation and recoupling conventions are those of Masbaum--Vogel \cite[Theorems~1--3 and Section~2]{MasbaumVogel1994}. Walker supplies the integral-weight and Lagrangian extension \cite[Definitions~1.1--1.6]{Walker1991}, with the Maslov-corrected mapping-cylinder convention of Gilmer--Masbaum \cite[Equations~(1)--(2) and Sections~9 and~11]{GilmerMasbaum2013}. The paper does not claim a new TQFT, skein module, recoupling theory, central extension, or geometric cube or pentagon. \subsection{Organization} Section~\ref{sec:standing-inputs} fixes the geometric, Walker, skein, and TQFT interfaces. Sections~\ref{trfb1odmscoc:sec-root-reduced-crosswalk}-- \ref{trfb1odmscoc:sec-squares-order-matrices-scope} construct the fixed comparison map and prove the ordinary operator theorem. Section~\ref{sec:complete-pentagon} constructs the compatible pentagon comparison package, proves the five-edge operator relation, and compares different auxiliary choices. The final section separates this fixed-level bridge from higher coherence and from the all-level program.