An Exact Algebraic Bifurcation in the Triangle-Plus-Center Central Configuration

We prove that the triangle-plus-center central configuration of the planar four-body problem with masses \((1, 1, 1, \mu)\) undergoes an exact stability transition at the critical mass ratio

\[\mu^* = \frac{81 + 64\sqrt{3}}{249},\]

the unique positive root of the irreducible quadratic \(249\mu^2 - 162\mu - 23 = 0\). At this algebraic mass ratio, the shape Hessian of the normalized potential \(\tilde{U}\) develops a degenerate eigenvalue doublet in the \(E\)-representation of the \(S_3\) symmetry group. For \(\mu < \mu^\) the configuration is a local minimum (Morse index 0); for \(\mu > \mu^\) it is a saddle (Morse index 2). This is the first explicitly computed degenerate central configuration at positive masses and disproves the universal non-degeneracy conjecture.

Despite the degeneracy, the critical point remains isolated: a Lyapunov-Schmidt reduction shows the reduced map has a nonzero quadratic term with resultant \(-238{,}963 \neq 0\), establishing isolation by degree theory. Breaking the \(S_3\) symmetry splits the doublet into two separate codimension-1 sheets of the degeneracy variety \(\mathcal{D}_4 \cap \mathbb{R}^4_{>0}\), proving it is a codimension-1 hypersurface—consistent with generic non-degeneracy (Albouy-Kaloshin) and CC finiteness (Smale's 6th Problem).

Keywords: central configurations, non-degeneracy, bifurcation, algebraic number, Smale's 6th Problem, four-body problem