Stacked Central Configurations with a Homogeneous Potential in R3

The paper’s Theorem 4 correctly classifies all 5-body stacked central configurations obtainable from 4-body ones for homogeneous potentials, with precise α-admissibility ranges and geometric constraints. The model’s core “equidistance” idea aligns with the paper’s necessary-and-sufficient stacking criteria, but it makes critical errors: (i) it incorrectly claims that the tetrahedral-plus-center case works for all α ≥ 2 (the paper requires α ∈ [3,∞) ∪ {2}), (ii) it misinterprets the admissible set Λ as merely encoding H ≥ 0, whereas Λ is a necessary condition for the base convex central configuration to exist, and (iii) it cites a finiteness paper as if it were a classification of spatial 4-body configurations. Consequently, the model’s solution overgeneralizes and conflicts with the paper on key parameter constraints.