\section{Introduction}
\label{p4:theorem-graph}

Start with a marked handlebody state and retain the relations imposed by its
chosen filling.  Before complementary capping, we want finite representatives
that can be read both in the disk topology and cyclotomically.  The obstacle
is already present in the coefficients: polynomial inclusion has no
continuous additive extension from the cyclotomic completion to the disk
power-series ring.  A common representative therefore cannot identify the
two completed targets.

This distinction also matters in the homological-block program.  Gukov,
Putrov, and Vafa introduced homological blocks through fivebrane
constructions \cite[Section~6.2]{GPPV1602}; Gukov, Pei, Putrov, and Vafa
formulated their reconstruction of WRT invariants and gave disk-series
formulas for negative-definite plumbings
\cite[Conjecture~2.1 and Appendix~A]{GPPV1701}.
Gukov--Manolescu then developed two-variable series for torus-boundary
manifolds \cite[Sections~5--7]{GukovManolescu}.
Here the object is an integral skein module before complementary capping.
Its common finite representatives organize disk and cyclotomic readings;
they do not identify that module with a homological block or turn a closed
cyclotomic invariant into a disk series.

The construction keeps the polynomial lattice in view.  At a finite stage,
the module Chinese remainder theorem separates the disk and cyclotomic
residues; compatible representatives then control the two inverse limits.
Regulated endpoint lattices make the pairings and transport meaningful, and
raw and saturated slide quotients retain the distinction between integral
relations and their generic saturation.  The one-holed-torus example follows
this construction toward the rootwise pairing and caps of the companion
paper on integral root fibers and closed factorizations
\cite{IntegralRootFibresClosedFactorizations2026}.

The simultaneous disk-adic and cyclotomic product-ideal filtration is
\(A^{2N}(q;q)_N\).

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[
  node distance=9mm and 12mm,
  box/.style={draw,rounded corners,align=center,inner sep=5pt,text width=30mm},
  wide/.style={draw,rounded corners,align=center,inner sep=5pt,text width=66mm},
  arr/.style={-{Latex[length=2mm]},thick}
]
\node[wide] (boundary) {one-holed-torus boundary lattice \(M\)\\before the complementary cap};
\node[wide,below=of boundary] (crt)
  {finite-stage joint quotient\\
   \(\begin{gathered}M/\mathfrak j_NM\cong\\
   M/A^{2N}M\times M/P_NM\end{gathered}\)};
\node[box,below=of crt,xshift=-24mm] (disk) {disk reading\\modulo \(A^{2N}\)};
\node[box,below=of crt,xshift=24mm] (root) {reduced root-of-unity fiber\\\(\mathcal S^+_{p}\)};
\node[box,below=of root] (pairing)
  {pairing-induced map\\\(\mathcal S_p=D_p^{-1}H_p\)};
\node[wide,below=of pairing] (caps)
  {identity and complementary caps\\
   \(y_p=e_0^{\mathsf T}\), \(h_p=y_pH_p\)};
\draw[arr] (boundary) -- node[right,align=left]{retain one finite\\representative} (crt);
\draw[arr] (crt) -- (disk);
\draw[arr] (crt) -- (root);
\draw[arr] (root) -- (pairing);
\draw[arr] (pairing) -- (caps);
\draw[arr] (root.south west) -- ++(-25mm,-2mm) |-
  node[pos=.25,left,align=left]{complementary\\capping} (caps.west);
\end{tikzpicture}
\caption{The one-holed-torus running example.  This paper constructs the
upper finite-stage and lattice data; the companion rootwise construction
gives the pairing and cap outputs.  The reduced root fiber is obtained by its
separate quotient-first construction, not by evaluating the whole completion.
The scalar caps do not reconstruct an uncapped state.}
\label{p4:fig:one-holed-torus-running}
\end{figure}

\subsection{Main results}

The all-genus filling quotient and generic spine use the attributed input in
Theorems~\ref{p4:retained:thm:c08-all-genus-quotient}
and~\ref{p4:retained:thm:c11-global-orthogonality}.  The module CRT and split completion,
Theorems~\ref{p4:retained:thm:c10-module-crt}
and~\ref{p4:retained:thm:c10-split-completion}, keep disk and cyclotomic
coordinates of one finite representative together without identifying the
completed targets.

The first lollipop norm and regulated lattice are fixed in
Theorems~\ref{p4:retained:thm:c11-norm}
and~\ref{p4:retained:thm:c11-regulated}.  Recoupling and separating transport
use the local-system and common-bridge statements
\ref{p4:retained:thm:c14-local-system}
and~\ref{p4:retained:thm:c15-bridge-common}--\ref{p4:retained:thm:c15-bridge-chamber}, with their marking, valuation,
and coefficient hypotheses.

Theorem~\ref{p4:retained:thm:c17-common-criterion} fixes the common-approximant
criterion; Theorem~\ref{p4:retained:thm:c17-no-map} proves the obstruction
above.  Theorem~\ref{p4:retained:thm:c17-hopf-cap} specifies the cap domain,
not a map on the whole completion.

\subsection{Local terminology introduced in this paper}

A \emph{proper height} is a height function with finite sublevel sets.  The
adjective \emph{disk--cyclotomic} abbreviates the simultaneous filtration
described above.  A \emph{lollipop lattice} is the regulated polynomial
lattice constructed here from the established lollipop basis.  Its Newton
interpolation products and regulator are specified in
Section~\ref{p4:retained:ch:lollipop-lattices}; they belong to this
construction, not to the name of the basis.  We use \emph{cotest} as a short name for a transported test functional.

\subsection{Inputs to the rootwise theory}

\begin{theorem}[Boundary constructions for the rootwise theory]
\label{p4a:thm:boundary-contract}
With the genus, marking, coefficient, endpoint, valuation, and cap-domain
hypotheses of the cited results, the following data are available to the
rootwise construction:
\begin{enumerate}
\item the marked filling quotient of
Theorem~\ref{p4:retained:thm:c08-all-genus-quotient};
\item the finite-stage joint quotient and split completion of
Theorems~\ref{p4:retained:thm:c10-module-crt}
and~\ref{p4:retained:thm:c10-split-completion};
\item the regulated endpoint lattices and recoupling transport of
Theorems~\ref{p4:retained:thm:c11-regulated}
and~\ref{p4:retained:thm:c14-local-system};
\item the separating bridge on its common domain from
Theorems~\ref{p4:retained:thm:c15-bridge-common}
and~\ref{p4:retained:thm:c15-bridge-chamber};
\item the raw and saturated towers and common-approximant criterion of
Theorems~\ref{p4:retained:thm:c16-towers}
and~\ref{p4:retained:thm:c17-common-criterion}; and
\item the no-map boundary and exact cap domain of
Theorems~\ref{p4:retained:thm:c17-no-map}
and~\ref{p4:retained:thm:c17-hopf-cap}.
\end{enumerate}
No clause identifies the disk and cyclotomic completions, evaluates the whole
completion at a root, or reconstructs an uncapped state from a cap scalar.
\end{theorem}

\begin{proof}
The six clauses collect the cited theorems without changing their
hypotheses or conclusions.  The final sentence is exactly their common
domain restriction.
\end{proof}

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[
  node distance=12mm and 5mm,
  box/.style={draw,rounded corners,align=center,inner sep=5pt,text width=32mm},
  note/.style={font=\small,fill=white,inner sep=2pt},
  arr/.style={-{Latex[length=2mm]},thick}
]
\node[box] (source) {marked Heegaard and skein source};
\node[box,right=of source] (finite) {filling quotient and finite-stage joint data};
\node[box,right=of finite] (lattice) {endpoint lattices, transport, and towers};
\node[box,below=of lattice] (p4b) {rootwise pairing, Rees, and cap comparisons};
\node[box,left=of p4b] (closed) {branch-specific closed scalars};
\draw[arr] (source) -- (finite);
\draw[arr] (finite) -- (lattice);
\draw[arr] (lattice) -- node[left,note]{stated domains and hypotheses} (p4b);
\draw[arr] (p4b.south) -- ++(0,-.7) -|
  node[pos=.25,below=3pt,note]{declared cap only} (closed.south);
\end{tikzpicture}
\caption{From marked boundary modules to rootwise comparisons.  This paper
constructs the finite-stage data and lattices; the companion rootwise
construction gives the pairing and capped comparisons.  The connecting
arrow retains the exact hypotheses and domains of the cited results; it
does not define a universal boundary state space.}
\label{p4:fig:global-architecture}
\end{figure}

\Needspace{4\baselineskip}
\begin{scopefirewall}[Global non-identifications]
\label{p4:fw:global}
This paper does not construct an untyped prime-independent all-root boundary
state, a root-specialization map on the whole Rees completion, or a global
root evaluation on the generic localized Habiro target.  It does not identify
Taylor, operational, stated-skein, Rees, or GNS boundary targets.  A
closed-cap-induced quotient is not promoted to a universal boundary
equivalence.  The general non-IHS block vector, complete $2$-primary sector,
positive-$b_1$ topological assignment, and refined arbitrary-genus theory
remain outside the proved boundary program.
\end{scopefirewall}