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We present a computationally efficient algorithm that is suitable for graphicprocessing unit implementation. This algorithm enables the identification ofall weak pseudo-manifolds that meet specific facet conditions, drawn from agiven input set. We employ this approach to enumerate toric colorable seeds.Consequently, we achieve a comprehensive characterization of$(n-1)$-dimensional PL spheres with $n+4$ vertices that possess a maximalBuchstaber number.  A primary focus of this research is the fundamental categorization ofnon-singular complete toric varieties of Picard number $4$. This classificationserves as a valuable tool for addressing questions related to toric manifoldsof Picard number $4$. Notably, we have determined which of these manifoldssatisfy equality within an inequality regarding the number of minimalcomponents in their rational curve space. This addresses a question posed byChen, Fu, and Hwang in 2014 for this specific case.

The characterization of $(n-1)$-spheres with $n+4$ vertices having maximal Buchstaber number