  We prove that the theory of the models constructible using finitely many
cofinality quantifiers - $C_{\lambda_{1},...,\lambda_{n}}^{*}$ and
$C_{<\lambda_{1},...,<\lambda_{n}}^{*}$ for $\lambda_{1},...,\lambda_{n}$
regular cardinals - is set-forcing absolute under the assumption of class many
Woodin cardinals, and is independent of the regular cardinals used. Towards
this goal we prove some properties of the generic embedding induced from the
stationary tower restricted to $<\mu$-closed sets.
