  We improve the state-of-the-art proof techniques for realizing various
spectra of $\mathfrak{a}_{\text{T}}$ in order to realize arbitrarily large
spectra. Thus, we make significant progress in addressing a question posed by
Brian in his recent work. As a by-product, we obtain many complete subforcings
and an algebraic analysis of the automorphisms of the forcing which adds a
witness for the spectrum of $\mathfrak{a}_{\text{T}}$ of desired size.
