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\title[{\protect\parbox{0.9\textwidth}{\protect\centering Phase-Labeled Doubles and Disjoint Handleslide Cubes with Complete Markings}}]{Phase-Labeled Doubles and Disjoint Handleslide Cubes with Complete Markings}
\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{25 August 2026}

\input{ams-metadata}

\begin{document}

\begin{abstract}
We prove two exact coherence results for marked handles in Walker's extended cobordism category.  At fixed disjoint sites, a matching pair of forward and
reverse phases contributes one local \(S^1\times S^2\) summand, whereas a
mismatch contributes an identity cylinder.  Thus the resulting double depends
only on the total Walker weight and the number of phase matches, not on the
orders of attachment.  For positive handleslides supported in pairwise-disjoint
protected sectors, we construct the complete marked Boolean cube and prove
that every two-face commutes with weight zero; consequently every finite route
has the same composite and the endpoint identities telescope.  An optional
odd-prime \(\mathrm{SO}(3)\)-TQFT corollary evaluates the fixed-site double as
\(\kappa_p^N(D_p^{\mathrm{GM}})^b\) times the identity.  Overlapping slides and
their pentagon are treated by a separate companion theorem.
\end{abstract}

\maketitle

\input{sections/00_main_results}
\input{sections/01_common_walker_data}
\input{sections/02_fixed_local_calculus}
\input{sections/03_fixed_order_traces}
\input{sections/04_fixed_match_dagger}
\input{sections/05_slide_setup}
\input{sections/06_slide_vertices_homology}
\input{sections/07_slide_covariance}
\input{sections/08_slide_cube}
\input{sections/09_ordinary_scalar}
\input{sections/10_scope_boundary}

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