  We construct a family of subdistributions of the Levi core
$\mathfrak{C}(\mathcal{N})$ called modified Levi cores
$\{\mathcal{M}\mathfrak{C}_{\mathcal{A}}\}_{\mathcal{A}}$ indexed over closed
distributions $\mathcal{A}$ that contain the Levi null distribution
$\mathcal{N}$ and are contained in the complex tangent bundle $T^{1, 0}b\Omega$
of a smooth bounded pseudoconvex domain $\Omega$. We show that Catlin's
Property ($P$) holds on $b\Omega$ if and only if Property ($P$) holds on the
support of one, and hence all, of the modified Levi cores. In $\mathbb{C}^2$,
all of the modified Levi cores coincide. For a smooth bounded pseudoconvex
complete Hartogs domain in $\mathbb{C}^2$ that satisfies Property ($P$), we
show that its modified Levi core is trivial. This contrasts with
$\mathfrak{C}(\mathcal{N})$, which can be nontrivial for such domains.
