The alpha spectrum of K-polystable toric $\mathbb{Q}$-Fano varieties

We explicitly determine the numerical spectrum of the ordinary, non-equivariant global invariant for $n$-dimensional K-polystable toric $\mathbb{Q}$-Fano varieties. Specifically, we prove that $$\{\alpha (X)\mid X\text{ is an }n\text{-dimensional K-polystable toric }\mathbb{Q}\text{-Fano variety}\}=\mathbb{Q}\cap [\frac{1}{n+1},\frac{1}{2}]\mathrm{.}$$ This result establishes a complete toric realization theorem, which strengthens a question raised by Liu and Zhuang and refines the recent construction results of Liu and Zhu. The initial construction of the examples in this work was suggested by GPT-5.6 Sol, and was subsequently refined and rigorously developed by the author.