Ultrafilter characters and local $\pi$-bases

We completely resolve Question~5.3 of Kania’s paper \cite{Kania} and establish a more general realization theorem. We show that for infinite cardinals $\mu ⩽\nu <\kappa <\lambda$, a space with $$\text{pc}(X)=\mu ,\text{pcl}(X)=\nu ,\text{pcc}(X)=\kappa ,\text{pcd}(X)=\lambda$$ exists if and only if $\kappa ⩽2^{\mu }$. The witnesses may be zero-dimensional and Tychonoff spaces, of the least possible cardinality $\lambda$. They may have the least possible number of non-isolated points, or alternatively be crowded. The main ingredient is an exact computation of the centered local $\pi$-character of star filter spaces in terms of ultrafilter character, obtained from a finite selection lemma for filter pushforwards. We also construct a single space with countable $r$-linked local $\pi$-bases for every finite $r$, centered value ${\omega }_1$, and any prescribed decreasing value greater than ${\omega }_1$.