D02 — G₂ 7-Body Choreography in ℝ⁷
| Field | Value |
|---|---|
| Domain | Dynamical Systems |
| System | Equal-mass 7-body gravitational problem in ℝ⁷ |
| Group | G₂ ⊂ SO(7) |
| H^k tier | H² |
| ISA | Forge (β ≈ β*) |
| Status | Conjectured |
| Opcodes | ORBIT · TWIST · BIND |
| Paper | Paper 552 |
Physical system
A conjectured choreography of 7 equal masses in ℝ⁷ whose symmetry group is the exceptional Lie group G₂ — the automorphism group of the octonions and the stabiliser of the Fano plane. The 7 bodies would occupy the 7 vertices of the Fano plane in the initial configuration, with the G₂ 3-form φ_{ijk} encoding which triples avoid collision.
This is the H² apex of the choreography ISA ladder: figure-eight (H⁰, ℝ²) → Lagrange triangle (H¹, ℝ³) → G₂ choreography (H², ℝ⁷).
Target category
Symp(G₂) — the category of G₂-structured symplectic manifolds. The relevant phase space is the G₂-invariant submanifold of T*(ℝ⁷)⁷ cut out by the associative 3-form φ.
Interpretation functor
F: C → Symp(G₂) defined by:
| Opcode | F(opcode) |
|---|---|
| ORBIT | Closed 7-body trajectory with G₂ winding number; period T_G₂ |
| TWIST | Berry phase accumulated around each body’s sub-orbit in ℝ⁷ |
| BIND | G₂ 3-form φ_{ijk}: encodes which triples (i,j,k) are Fano lines — collision-avoidance constraint |
ISA programme
INIT: LABEL[q₁…q₇ | Fano vertex IC] -- 7 bodies at Fano vertices
FANO: BIND[φ_{ijk} | (i,j,k) ∈ Fano lines]-- G₂ 3-form collision constraint
FLOW: ORBIT[Φ_t : T*ℝ⁷^7 → T*ℝ⁷^7] -- Hamiltonian flow
TWIST: TWIST[Berry phase per sub-orbit] -- accumulate H¹ holonomy
CLOSE: ORBIT[q(T) = σ(q(0))] -- close up to G₂ permutation σ
PERIOD: LABEL[T_G₂] -- period eigenvalue (unknown)
Computable output
- Conjectured period T_G₂: unknown — the primary output of experiment x552d.
- G₂ winding structure: 7 sub-orbits related by the 7-fold symmetry of the Fano plane. Each body traverses the same curve; consecutive bodies are offset by T_G₂/7.
- BIND closure: the G₂ 3-form φ_{ijk} evaluated on each Fano triple must equal the associator of the octonion units e_i, e_j, e_k. This is the collision-avoidance condition — bodies on the same Fano line repel by the non-associativity of ℝ⁷.
- H² necessity: G₂ is the minimal exceptional group. Its 3-form φ is a non-trivial H² class in H²(G₂/SO(4)) — there is no way to encode the 7-body Fano constraint using only ORBIT (H⁰) or TWIST (H¹). BIND is required.
Why this matters
The figure-eight (D01) requires no group theory — any 3 equal masses in ℝ² find it. The G₂ choreography would be the first exceptional choreography: one that exists only because ℝ⁷ admits an exceptional geometry (the G₂ structure) not present in any other dimension. It would prove that the H² tier of the ISA is physically realised in classical mechanics, not just quantum computing.
Validation status
- x552d (SHA c50f328): first numerical search for a 7-body G₂ choreography in ℝ⁷. Convergence behaviour documented; existence not yet confirmed.
- x552e: 4-body sanity check (proposed) — verify that a known 4-body choreography in ℝ⁴ is recovered before extending to 7-body ℝ⁷.
- Existence would follow from a G₂-equivariant version of the Chenciner-Montgomery variational argument — an open problem in symplectic geometry.