  We introduce a new package of Floer data of $\lambda$-sectorial almost
complex structures $J$ and sectorial Hamiltonians $H$ on the Liouville sectors
introduced by Ganatra-Pardon-Shende the pairs of which are amenable to the
maximum principle for the analysis of pseudoholomorphic curves relevant to the
studies of wrapped Fukaya categories and of symplectic cohomology. It is also
amenable to the strong maximum principle in addition when paired with
cylindrical Lagrangian boundary conditions. The present work answers to a
question raised by Ganatra-Pardon-Shende concerning a characterization of
almost complex structures and Hamiltonians in that all the relevant confinement
results in the studies of wrapped Fukaya category, symplectic cohomology and
closed-open (and open-closed) maps between them can be uniformly established
via the maximum principle through tensorial calculations, Hamiltonian calculus
and sign considerations without making any estimates. Along the way, we prove
the existence of a pseudoconvex pair $(\psi,J)$ such that \emph{$J$ is
$d\lambda$-tame and $\psi$ is an exhaustion function of
$\operatorname{nbhd}(\partial_\infty M \cup \partial M)$ that also satisfies
the equation $-d\psi \circ J = \lambda$ thereon for any Liouville sector with
corners $(M,\lambda)$}.
