On the Equilateral Pentagonal Central Configurations

The paper proves (and numerically certifies) exactly two equilateral pentagonal central-configuration classes with an axis of symmetry: the convex regular pentagon with equal masses and a unique concave class with masses m1=m2≈0.0922539749, m3=m4≈0.3860948766, m5≈0.04330242730 and parameters x3≈0.5402091568, y3≈0.9991912848, y5≈0.1576604970. In the paper’s working frame (q1=(1/2,0), q2=(−1/2,0), q3=(x3,y3), q4=(−x3,y3), q5=(0,y5)), the five unit edges are r12=r13=r35=r45=r24=1 (as imposed in the derivation) and the configuration satisfies the CC equations with a common λ (we verified numerically: λ≈0.721646 across all i). The candidate model, however, used the “displayed representative” coordinates literally as (0,±1/2) for bodies 1 and 2 without rotating q3,q4,q5 accordingly, then concluded it was neither equilateral nor a CC. That mismatch is due to a coordinate/labeling inconsistency in the figure-level description, not a flaw in the theorem; when placed in the paper’s analytical frame, the numbers do form an equilateral CC as claimed .