\documentclass[11pt,a4paper]{amsart}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm,mathrsfs,mathtools,microtype,booktabs,array}
\usepackage[margin=1.02in]{geometry}
\usepackage[svgnames]{xcolor}
\input{preamble}

% Bind shared-counter reference classes locally, independently of automatic
% amsthm detection. Numbers, labels, and printed environment names are unchanged.
\AddToHook{env/proposition/begin}{\crefalias{theorem}{proposition}}
\AddToHook{env/lemma/begin}{\crefalias{theorem}{lemma}}
\AddToHook{env/corollary/begin}{\crefalias{theorem}{corollary}}
\AddToHook{env/claim/begin}{\crefalias{theorem}{claim}}
\AddToHook{env/definition/begin}{\crefalias{theorem}{definition}}
\AddToHook{env/construction/begin}{\crefalias{theorem}{construction}}
\AddToHook{env/convention/begin}{\crefalias{theorem}{convention}}
\AddToHook{env/remark/begin}{\crefalias{theorem}{remark}}
\AddToHook{env/warning/begin}{\crefalias{theorem}{warning}}

\hypersetup{
  pdftitle={Marked Positive Handleslide Pentagons in Compression Bodies: Relative Coherence},
  pdfauthor={Hao-Yu Sun},
  pdfsubject={Relative coherence for marked positive handleslide pentagons in compression bodies}
}

\title[{\protect\parbox{0.9\textwidth}{\protect\centering Marked Positive Handleslide Pentagons in Compression Bodies:  Relative Coherence}}]{Marked Positive Handleslide Pentagons in Compression Bodies:\\
Relative Coherence}
\author{Hao-Yu Sun}
\address{Weinberg Institute, The University of Texas at Austin, 2515 Speedway, C1600, Austin, Texas 78712, USA}
\address{SpaceXAI, USA}
\email{hkdavidsun@utexas.edu}
\date{23 August 2026}

\input{ams-metadata}

\begin{document}
\begin{abstract}
Let \(W:X\to Y_{\mathrm{geom}}\) be a completed oriented compression body
with \(N\geq 3\) ordered product \(1\)-handles and a supplied protected
three-handle sector.  We realize one positive handleslide pentagon by five
diffeomorphisms with compatible complete endpoint markings.  An isotopy fixed
on the entire protected boundary collar identifies the two-edge and
three-edge route composites, so the comparison takes place at the level of
marked maps rather than homology matrices.

The construction fixes the marked disk systems at all five vertices before
choosing any edge map, so incident edges use the same endpoint data.  The
cut-ball maps intertwine all fifteen disk pairings on common full-normal
intervals and therefore descend smoothly through the paired faces.  Extending
these quotient maps by the identity gives global representatives.  The same
maps satisfy the finite compatibility conditions, including at \(N=3\), when
the unused-handle domains are empty but remain part of the typed construction.

The resulting boundary maps preserve the embedded meridian and
product-framed dual-longitude rows, so their homological actions have blocks
\(M_e\) and \(M_e^{-T}\) with no mixed shear.  They also preserve the same
rational disk-kernel Lagrangian.  Walker's forward multiplication law lifts
the relative isotopy to a literal weight-zero equality of mapping cylinders, including the seam and route weights.
\end{abstract}

\maketitle

\input{sections/01_introduction}
\input{sections/02_protected_sectors}
\input{sections/03_five_edge_pentagon}
\input{sections/04_main_theorem}
\input{sections/05_vertex_geometry}
\input{sections/06_endpoint_atlases}
\input{sections/07_naturalization}
\input{sections/08_relative_extensions}
\input{sections/09_edge_pairing_stabilization}
\input{sections/10_descent_global}
\input{sections/11_pc5}
\input{sections/12_homology}
\input{sections/13_rigidity_pentagon}
\input{sections/14_walker}
\input{sections/15_scope_boundary}

\section*{Acknowledgments}

The initial exploration, analytical derivations, and drafting of this manuscript were substantially assisted by OpenAI's GPT-5.6 Sol and GPT-6 Astra under the author' direction. The author have checked the arguments and take responsibility for the mathematical claims.

\appendix
\input{appendices/a_cut_schedules}
\input{appendices/b_collars_all_jets}
\setcounter{section}{5}
\input{appendices/f_local_positive_slide}

\bibliographystyle{alpha}
\bibliography{references}

\end{document}