Realizing arbitrarily large spectra of ${\mathfrak{a}}_{\text{T}}$

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.