A stochastic optimization algorithm for analyzing planar central and balanced configurations in the n-body problem

The paper defines BC(S) and CC precisely, formulates the isotropy-weighted Morse identity Σx (−1)^{h(x)}/i(x) = (−1)^n/(n(n−1)) (its eq. (34)), and uses H = D^2U_n + U_n S M to compute Morse indices and isotropy classes, then reports counts Nsol for n = 3,…,12 as 2, 4, 5, 9, 14, 20, 42, 67, 114, 191, respectively; it also identifies three additional n = 10 CCs beyond Ferrario’s 64 (Fig. 1, Table 2), and exhibits balanced configurations without any axis of symmetry for n = 4 and n = 10 (Figs. 11–12). All of these match the candidate solution’s counts, Morse-identity statement, and qualitative claims. The paper also acknowledges that completeness is inferred numerically and via the Morse identity under a Morse-function assumption and Krawczyk certificates, not by a formal exhaustive proof; the candidate solution similarly relies on these assumptions. Within that shared framework, the claims agree, hence the verdict.