\section{Introduction} \label{sec:introduction} Consider a positive three-disk handleslide pentagon in a completed compression body. On homology, its two paths are represented by products of elementary matrices, and the two products agree. Does this agreement lift to equality of fully marked relative mapping cylinders? The arithmetic alone does not supply the actual diffeomorphisms, compatible endpoint markings, or relative isotopy needed to compare the composites. For one positive pentagon in a supplied standard three-handle sector of a completed compression body, we construct a two-edge route and a three-edge route whose composites are isotopic relative to the entire protected marking. The resulting marked mapping cylinders satisfy a literal weight-zero Walker equality. The construction retains signed cut faces with full normal collars and embedded meridian and dual-longitude rows. Their matrix identity is an exact integral coordinate check for the geometric relation. The handleslide pentagon itself is known: Hatcher--Thurston cut-system relations contain pentagon cells \cite{HatcherThurston1980}, and the Heegaard-diagram complex of Juh\'{a}sz--Thurston--Zemke contains the directly relevant three-curve handleslide pentagon \cite[arXiv~v4, Appendix~A, Definition~A.3\textup{(6)} and Figure~22]{JuhaszThurstonZemke2021}. The priority-sensitive content here is only the printed refinement---one supplied protected sector, coordinated selected representatives, complete retained markings, an isotopy fixed on the full protected collar, and the resulting literal Walker relation. Without those qualifiers, the headline reduces to known move-complex coherence. In the sutured compression-body setting, Qin proves simple connectivity after attaching six kinds of triangle, square, and pentagon cells to a complex of isotopy classes of cut-systems \cite[Theorem~1.1, Definitions~2.5 and~4.2, and the opening of \S4]{Qin2025}. Those vertices are isotopy classes, whereas here the selected representatives carry the prescribed full-collar data. The full-collar isotopy of the selected complete-marked routes is the narrower question addressed here. \subsection{Coordinating the geometric representatives} The pentagon has five vertices and five directed edges: a two-edge route and a three-edge route with common endpoints. It is tempting to realize each edge separately by a standard handleslide map and compare the route matrices afterward. That procedure leaves three gaps. First, an endpoint shared by two edges can acquire two unrelated cut-ball charts, collar depths, or boundary parameterizations. Equality of endpoint surfaces then does not mean equality of the complete endpoint germs needed for composition. Second, agreement of a boundary map with disk pairings at normal coordinate zero is insufficient for smooth quotient descent: both directed pairing identities must hold on one full-normal interval, with all jets. Third, the cut-ball map, its quotient descent, and its global extension have different domains and roles. In particular, the full two-sided collar of the exterior attachment disk exists only after descent and global extension; it cannot be assigned retroactively to a cut-level map. The proof resolves all three gaps by reversing the order of construction. All five vertices are built first, with one common positive collar depth and one literal collection of endpoint data at every shared endpoint. Only then are the ten incidence-labelled endpoint markings and charts and the five edge representatives selected. This temporal order is part of the theorem, rather than an organizational preference. \subsection{From the supplied sector to the Walker relation} Fix \(g\geq1\), \(N\geq3\), a completed oriented compression body \(W:X\to Y_{\mathrm{geom}}\), three ordered distinguished handles, an exterior completed row, and the fully protected standard sector specified in \cref{def:supplied-protected-sector}. The sector includes literal product coordinates, positive carrier push-offs, three compatible zero-framed band germs, simultaneous smoothing data, reserve regions, and one common exterior parameterization. Thus the theorem does not search an arbitrary compression body for standard position. The proof of \cref{thm:complete-marked-pentagon} begins with five independently constructed vertex disk systems and their cut balls. Ten endpoint markings and charts are coordinated before the five cut maps \(\widetilde F_e\) are chosen. Each cut map carries the actual face, ribbon, spoke, detector, meridian, and dual-longitude data and satisfies all fifteen full-normal pairing identities. These identities let every \(\widetilde F_e\) descend to a smooth map \(f_{e,U}\) of the active quotient sector and then extend by the coordinate identity to a global map \(F_e\) of \(W\). The construction keeps these three stages distinct, so a cut-level assertion is not confused with a global one. The same maps satisfy the finite compatibility conditions, including the five unused-handle records whose domains are empty when \(N=3\). Their geometric proof is given in \cref{sec:pc5}. % Reader-reference migration: \cref{app:pc5-census} remains an internal alias % for the repository-retained finite table omitted from the public driver. Because the maps transport embedded meridian and product-framed dual-longitude rows before homology is read off, their active homology blocks are \(M_e\) and \(M_e^{-T}\), with both mixed blocks zero. Every map and its inverse preserve the same rational disk-kernel Lagrangian. A cut-level rigidity path is then kept fixed on a full protected collar, descended through the pairings for every parameter value, and extended globally. It gives a relative isotopy between the two route composites. The relative isotopy first yields equality of the marked mapping classes. Walker's forward composition law is then applied in its Lagrangian-weighted form \cite{GilmerMasbaum2013}. Since the common Lagrangian is preserved in both directions, each ordered seam Maslov index is zero, so the two products of weight-zero mapping cylinders are literally equal. The local analytic-topological ingredients are used with their relative conditions exposed. Relative smoothing of a doubled three-manifold homeomorphism uses the classical relative homeomorphism-approximation theorem, in the form stated by Pardon \cite[arXiv~v2, Theorem~4.6]{Pardon2020}; the rel-boundary ball rigidity step uses the Smale-conjecture consequence proved by Hatcher \cite{Hatcher1983}; and the passage from fixed boundary and normal-tube data to a smooth fixed-germ isotopy uses Cerf's relative results \cite{Cerf1961}. These results enter at specifically marked steps. They do not replace the explicit endpoint, pairing, or collar constructions. \subsection{Homology as an exact coordinate check} Let \(S_v\) record the three meridian descendants at a vertex and put \(M_e=S_wS_v^{-1}\) for an edge \(e:v\to w\). Direct multiplication gives the same matrix along the two-edge and three-edge routes, and the inverse-transpose products agree as well. We record this calculation early because it is a useful typing check: for example, the two edges involving the letter \(B\) have different source roles and different matrices. We do not use the common product to infer a mapping-class relation. A homological representation need not detect the geometry with complete marking, the required full-collar behavior, or a relative isotopy. In the proof, the direction of implication is therefore \[ \begin{gathered} \text{embedded marked geometry} \longrightarrow \text{relative isotopy}\\ {}\longrightarrow \text{mapping-class equality} \longrightarrow \text{weight-zero Walker equality}, \end{gathered} \] while the matrix arithmetic remains an independent exact check. \subsection{Exact scope} For every supplied protected sector the theorem constructs one coordinated system. It asserts neither uniqueness nor relations among several pentagons, and it does not by itself imply stabilization, global Heegaard invariance, or an operator-level comparison. The precise boundary is stated in \cref{sec:scope-boundary}. \subsection{Organization} Sections~\ref{sec:protected-sectors}--\ref{sec:main-theorem} fix the input, type the pentagon, and state the witness theorem. The middle sections follow the construction diagram: vertex data, endpoint charts, relative extensions, cut maps, descent, compatibility, and homology. Sections \ref{sec:rigidity-pentagon} and~\ref{sec:walker} prove the relative isotopy and weight-zero Walker equality. The mathematical appendices retain the local cut, collar, and positive-slide arguments needed by those proofs.