\section{Boundary results and limitations}
\label{p4a:ch:interface}

The constructions developed here provide marked source modules, filling
quotients, finite-stage joint quotients, endpoint lattices, exact recoupling and
separating transport, raw and saturated quotient towers, and the declared
cap domains.  They furnish the inputs to the companion paper on integral
root fibers, Rees covariance, caps, and sewing, subject to the boundary
constructions in Theorem~\ref{p4a:thm:boundary-contract} and the hypotheses of
the individual results.

These results do not define root specialization on an entire completion,
identify the disk and cyclotomic completions, or provide an inverse operation
that reconstructs a boundary state from a capped scalar.  The limitations
are structural: the relevant source and target objects lie in different
domains.