\section{Boundary outputs and limits}
\label{p4:ch:interface}

The constructions give maps with specified domains and closed scalars, not one universal
boundary state space.  They comprise the integral pairing image, factorial/Rees
covariance, the strict rootwise/Rees object and theta endpoint, the permitted
line-valued and scalar contractions, and the exact obstructions to extending
those constructions.  The perturbative, Habiro, operational, and
stated-skein/Morita branches retain their separate source types and are
compared only after a declared cap or sewing map.

The companion paper on Heegaard realizations and radial comparisons with
$\widehat Z$
\cite{ClosedHeegaardDescentRadialZhat2026} uses the transported integral
lattice and Habiro cap in the stated integer-homology-sphere domain, together
with its root and Taylor coordinates.  This construction does not identify the factor
lattices, evaluate a generic boundary element at every root, or reconstruct a
boundary state from the resulting closed scalar.

\begin{scopefirewall}[Conditions on applications]
Each application retains the target type, genus and root range,
chamber or variance convention, cap or scalarization step, and any IHS or
primitive-cap hypothesis of the result used.  Omitting those qualifiers
changes the claim.
\end{scopefirewall}