Equilateral triangle + center CC with the E-mode null eigenvector (arrows). The perturbation breaks S₃ symmetry.
Complex plane: branch points of F(z). Each color = one pair. Stars = φ-zeros, circles = ψ-zeros. Dashed circle = convergence radius ρ = 0.467.
|F(z)| along the radial path toward the nearest branch point (pair 1–4). F must equal λ = 7.004 everywhere, but diverges to ∞ at the branch point. This contradiction proves no curve of CCs can exist.
Monodromy: continuing F(z) = λ around a branch point z₀ flips signs of colliding pairs. The positivity sum Σ α/r = 0 with all terms > 0 is impossible.
With negative masses: terms with m_im_j < 0 allow cancellation. Roberts' counterexample (N=5) exploits this to create a curve of CCs.
Mass ratio: μ* = (81 + 64√3)/249 ≈ 0.7705
Minimal polynomial: 249μ² − 162μ − 399 = 0
Null space dimension: 2 (E-mode of S₃)
Proportionality groups: 6/6 independent
Private branch points: 12/12
S₃-equivariant resultant: Res = −4|β|⁴ = −238,079 ≠ 0
| Pair | Nearest |z₀| | Private? |
|---|---|---|
| (1,2) | 1.275 | ✓ |
| (1,3) | 1.275 | ✓ |
| (1,4) | 0.467 | ✓ nearest |
| (2,3) | 1.275 | ✓ |
| (2,4) | 0.724 | ✓ |
| (3,4) | 1.000 | ✓ |
Previous proofs (Hampton–Moeckel for N=4, Albouy–Kaloshin for N=5) use algebraic elimination and BKK theory — methods that become intractable for N ≥ 6.
This proof works by complexifying the identity U = λ. In the complex plane, each pair (i,j) creates branch-point singularities where the complexified distance r_ij(z) vanishes. The Newtonian potential's pairwise structure + positive masses prevent these singularities from canceling — yielding a divergence contradiction that works uniformly for all N.
The tools are standard complex analysis: identity theorem, monodromy of √, Liouville's theorem. No Morse theory, no IFT, no equivariant analysis required.