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% FixedLevelQuantumHandleslideCoherence2026
\title[{\protect\parbox{0.9\textwidth}{\protect\centering Fixed-Level TQFT Transport of Marked Handleslide Coherence:  Disjoint Cubes and a Three-Handle Pentagon}}]{Fixed-Level TQFT Transport of Marked Handleslide Coherence:\\
Disjoint Cubes and a Three-Handle Pentagon}
\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{24 August 2026}

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\begin{document}

\begin{abstract}
Fix an ordinary level $K\geq1$ and a fully specified family of pairwise disjoint positive handleslides on a compression body.  The main question is how a literal geometric relation becomes a literal finite-level operator relation.  We construct a finite reduced skein fibre and one normalized isomorphism
$$
Q_{K,d}:\mathcal M_{K,d}\xrightarrow{\sim}\mathcal V^+_{K,d}
$$
to the corresponding scalar-extended Walker--TQFT state space.  Reusing this map at every vertex sends the supplied weight-zero slide cylinders to $E_K$-linear automorphisms: geometric squares become operator squares, finite slide orders give the same operator, and forward/reverse maps telescope for all declared phases and integer weights.  The construction is fibrewise and makes no comparison between levels.  These operator relations transport the geometric disjoint-cube and marked-pentagon relations established in companion works; they do not constitute new geometric coherence theorems.


We then treat a fully specified positive three-handle pentagon.  Its vertex-zero compression body admits a compatible fixed-level comparison core: a rational disk-kernel cap, a normalized complementary handlebody, a reduced spine model, and one universal-construction isomorphism $Q_{K,d}^{\mathfrak q}:\mathcal M_{K,d}^{\mathfrak q}\xrightarrow{\sim}\mathcal V^+_{K,d}$. The geometric weight-zero Walker pentagon consequently has a literal operator image over \(E_K\), with explicit inverse formulas and a fixed-basis matrix corollary.  Different comparison cores are related by a typed comparison groupoid and give intertwined operator systems; no canonical comparison or literal equality between different coordinate choices is asserted.
\end{abstract}

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\input{sections/04_squares_order_matrices}
\input{sections/05_pentagon_operator_image}
\input{sections/06_scope_and_outlook}

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