\section{Introduction}
\label{p4b:theorem-graph}

A marked boundary state can be read in a reduced root fiber and through a
formal Rees coordinate, but these readings have different domains.  We
compute the integral image of the map induced by the Hopf pairing and
construct a handlebody-covariant Rees bridge from common finite-stage
representatives.  The common representatives retain both readings without
making the completed Rees source root-valued: at an admissible odd prime,
no continuous homomorphism on that entire completion extends algebraic root
specialization to the reduced coefficient field.  The reduced root fiber
also remains distinct from the disk reading in the companion paper on
marked Heegaard boundary modules \cite{MarkedHeegaardBoundaryModules2026}.

The motivation for comparing these readings comes in part from homological
blocks: the fivebrane construction of Gukov--Putrov--Vafa
\cite[Section~6.2]{GPPV1602} and the WRT reconstruction proposed by
Gukov--Pei--Putrov--Vafa \cite[Conjecture~2.1]{GPPV1701} relate disk
series to root-of-unity invariants.  The torus-boundary series and gluing
formulas of Gukov--Manolescu \cite[Sections~5--7]{GukovManolescu} make the
boundary comparison particularly concrete.  The construction here instead
starts with integral Hopf pairings and reduced root fibers.  Its Rees
coordinate is formal, and its closed cyclotomic caps retain their stated
domains; none of these outputs is identified with a disk homological block.

Smith invariants in Newton interpolation coordinates compute the integral
pairing image in
Theorems~\ref{p4:thm:topological-riesz}
and~\ref{p4:thm:smith-image}, using attributed Hopf, Newton, and integral-TQFT
inputs.  Section~\ref{p4:chap:smith-newton} specifies the core variable,
nodes, and coefficient ring, and separates the polynomial basis from its
rescaled lattice frames.  Factorial denominator-clearing factors and compatible representatives
supply the covariance and endpoint construction in the first theorem below;
the second states the obstructions.

\subsection{Local terminology introduced in this paper}

The phrase \emph{rootwise/Rees state} is local shorthand for a state carried
by the common finite-stage representatives of the marked boundary-module
construction,
with its reduced root-of-unity fiber and Rees endpoint kept distinct.  A
\emph{reduced root-of-unity fiber} is the quotient-first specialization at an
admissible odd prime, after the root-level admissibility and negligible-class
relations stated in the cited theorem have been imposed.  Thus \emph{root
specialization} below never means evaluation of the whole Rees completion.
Likewise,
\emph{finite-prime-support data} means the explicitly finite set of root
fibers adjoined after closure; it is not a new field-standard structure.

The one-holed-torus diagram separates the disk reading from the rootwise path.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[
  node distance=9mm and 12mm,
  box/.style={draw,rounded corners,align=center,inner sep=5pt,text width=31mm},
  wide/.style={draw,rounded corners,align=center,inner sep=5pt,text width=44mm},
  arr/.style={-{Latex[length=2mm]},thick}
]
\node[wide] (contract) {finite-stage marked boundary data};
\node[box,below=of contract,xshift=-23mm] (root) {reduced root-of-unity fiber\\\(\mathcal S^+_p\)};
\node[box,below=of contract,xshift=23mm] (disk) {disk reading\\modulo \(A^{2N}\)};
\node[wide,below=of root] (pairing) {integral image of the map induced by the Hopf pairing};
\node[wide,below=of pairing] (strict) {strict rootwise/Rees module and theta endpoint};
\node[wide,below=of strict] (cap) {declared line-valued or scalar cap outputs};
\draw[arr] (contract) -- (root);
\draw[arr] (contract) -- (disk);
\draw[arr] (root) -- (pairing);
\draw[arr] (pairing) -- (strict);
\draw[arr] (strict) -- (cap);
\end{tikzpicture}
\caption{The rootwise path in the one-holed-torus running example.  The disk reading
remains a separate output of the finite-stage boundary construction; only the reduced
root fiber enters the pairing-image and rootwise path.  The last arrow is
defined only on its stated cap domain.}
\label{p4b:fig:one-holed-torus-running}
\end{figure}

\subsection{Main results}

\begin{theorem}[Rootwise/Rees construction and integral pairing image]
\label{p4b:thm:rootwise-rees-pairing}
In the root, genus, coefficient, and completion domains stated by the cited
results below, Smith invariants in Newton coordinates determine the saturated
image of the map induced by the Hopf pairing; the factorially cleared Rees
bridge is handlebody-covariant; and common finite-stage representatives form
the strict rootwise/Rees module with its theta endpoint.
\end{theorem}

\begin{proof}
The three clauses are respectively
Theorems~\ref{p4:thm:smith-image},
\ref{p4:thm:factorial-clearer}--\ref{p4:thm:handlebody-covariance}, and
\ref{p4:thm:common-approximant-carrier}--
\ref{p4:thm:theta-root-rees-endpoint}.  Each clause retains the domain and
target of its cited statement.
\end{proof}

The quantitative bridge uses the polynomial regulators and chart gauges
defined in Section~\ref{p4:chap:factorial-joint}.  Its rapid sector
requires every growth rate.  A fixed low-height core is chosen before the
requested output rate.  Ordinary polynomial vector approximation in
Theorem~\ref{p4:thm:handlebody-chambers} requires both transported output
families to lie in the rapid or exact-intersection sector.  Localized
finite-core data instead admit Laurent interpolation for prescribed disk
precision and finite scalar-root data, with the stated coefficient
localization and without an all-root coherence assertion.

\begin{theorem}[Principal root-specialization and state-lift obstructions]
\label{p4b:thm:principal-obstructions}
For each odd prime \(p\ge5\), no continuous homomorphism from all of
\(\mathcal R_\lambda\) to the reduced \(p\)-th coefficient field extends
\(A\mapsto A_p\).  Moreover, scalar Galois covariance
does not admit the state-natural lift to the strict rootwise/Rees module
excluded below.  These obstructions do not preclude the stated line-valued or
capped scalar outputs.
\end{theorem}

\begin{proof}
The extension obstruction is
Theorem~\ref{p4:thm:rees-no-evaluation}; the state-lift obstruction is
Theorem~\ref{p4:thm:galois-state-no-go}.  Their positive-domain limitations
are part of those statements and yield the final sentence.
\end{proof}

\subsection{Boundary inputs and closed comparisons}

The finite-stage inputs come from the marked boundary-module construction
\cite{MarkedHeegaardBoundaryModules2026},
Theorem~\ref{P4A-p4a:thm:boundary-contract}.  The table records their
hypotheses and domains.

\begin{center}
\small
\begin{tabular}{@{}p{0.34\textwidth}p{0.50\textwidth}@{}}
\toprule
\textbf{Imported object} & \textbf{Source result and hypotheses} \\
\midrule
Filling quotient and generic spine module &
Theorems~\ref{P4A-p4:retained:thm:c08-all-genus-quotient}
and~\ref{P4A-p4:retained:thm:c11-global-orthogonality}; retain their genus,
marking, and coefficient hypotheses. \\
Finite-stage joint quotient and split completion &
Theorems~\ref{P4A-p4:retained:thm:c10-module-crt}
and~\ref{P4A-p4:retained:thm:c10-split-completion}; no root specialization of the
whole completion. \\
Endpoint lattices and transport &
Theorems~\ref{P4A-p4:retained:thm:c11-regulated},
\ref{P4A-p4:retained:thm:c14-local-system}, and
\ref{P4A-p4:retained:thm:c15-bridge-common}--\ref{P4A-p4:retained:thm:c15-bridge-chamber}; retain endpoint and valuation tags. \\
Towers and cap domain &
Theorems~\ref{P4A-p4:retained:thm:c16-towers},
\ref{P4A-p4:retained:thm:c17-common-criterion},
\ref{P4A-p4:retained:thm:c17-no-map}, and
\ref{P4A-p4:retained:thm:c17-hopf-cap}; no disk/cyclotomic identification and no
uncapped-state reconstruction. \\
\bottomrule
\end{tabular}
\end{center}

Established stated-skein/Morita sewing and the generic Habiro cap, followed
by the cap-induced universal-construction quotient, give branch-specific
closed comparisons on their cited domains.  They neither identify uncapped
boundary modules nor provide new generic sewing theorems.