Symmetric relative equilibria with asymmetric positive masses in the Newtonian 1+N coorbital problem, for N = 10 and every even N ≥ 4

In 2022 Deng, Hampton and Wang proved that the Newtonian 1+N coorbital problem has relative equilibria whose configuration has an axis of symmetry while the positive masses do not, for N = 4, 6, 8. They conjectured that the same holds for larger even N, and the first open case is N = 10. For the Newtonian potential we prove the conjecture for every even N ≥ 4, by one argument that also recovers N = 4, 6, 8. The proof has two parts. The first is algebraic. Consider a configuration that is symmetric under a reflection fixing none of its points. On it, the mass matrix splits into a block A and the block −Aᵀ. Hence every relative equilibrium with positive symmetric masses is also a relative equilibrium for a two-dimensional cone of positive masses, almost all of which lack the symmetry. This also explains the Pfaffian computations of Deng, Hampton and Wang, since the Pfaffian equals ±det A. The second part is local. Through the regular N-gon passes a hypersurface of dimension N/2 − 1 of symmetric configurations that admit positive symmetric masses. Its generic points have exactly one nontrivial symmetry, the reflection. The nondegeneracy needed at the regular N-gon reduces to three conditions: one sign, which holds for every N; a family of trigonometric sums, positive by a concavity argument except the first; that first sum, which is nonzero for every even N (analytically for N ≥ 16, by certified ball arithmetic for N ≤ 14). For N = 10 we also give an explicit relative equilibrium with ten positive masses and a single axis of symmetry, certified by ball arithmetic. The algebraic part, the concavity lemma and the N = 10 sign inequality are verified in Lean 4 with Mathlib, with no sorry and only the standard axioms. The paper names each step not formalised and gives the reason.