Three Search Tricks for Equal-Mass Three-Body Choreographies via Fano Geometry

This paper introduces three search techniques for equal-mass three-body choreographies grounded in the Fano geometry of the three-body phase space, complementing the winding-number search of the companion paper (10.5281/zenodo.20369300).

Trick 1 — Symmetry reduction: the Fano orbit label L of a choreography's trajectory determines a fixed-point subspace of ℝ¹² that reduces the initial-condition search from twelve to two dimensions. Different orbit labels correspond to different symmetry subgroups, partitioning the search space by Fano valence.

Trick 2 — Frequency ratio targeting: the orbit label L predicts the rational frequency ratio ω₁/ω₂ = p_L/q_L at which a new choreography can bifurcate. Primary-orbit labels L ∈ {1,2,3} predict low-order resonances (1:2, 1:3, 2:3); secondary-orbit labels L ∈ {4,5,6} predict higher-order resonances.

Trick 3 — XOR-Fano bifurcation: the XOR-Fano composition rule χ_{L₁} ⊗ χ_{L₂} = χ_{L₁⊕L₂} predicts which orbit families can bifurcate from which. Starting from a known orbit of label L₁, new orbits of label L₂ are found by perturbing in the direction χ_{L₁⊕L₂}.

The PSL(2,7) conjecture is stated: within the (Ab)^{3k} family, stable choreographies exist if and only if k divides 168 = |PSL(2,7)|. Three cases (k=1,8,14) are confirmed; eight cases remain open pending high-precision computation.

The Fano boundary test is introduced: the Freudenthal determinant criterion |det_F| < 42.5 is tested against choreographies from the Šuvakov-Dmitrašinović catalogue. All well-closed orbits pass; failures have insufficient initial-condition precision and are inconclusive. The provisional result supports Fano universality across gravitational, nuclear, and quantum computing domains.

Keywords

Three-Body Problem, Choreographies, Fano Geometry, Fano Orbit Label, XOR-Fano Rule, PSL(2,7), Freudenthal Determinant, KAM Theory, Symplectic Integration, Frequency Ratio, Bifurcation, Equal-Mass Orbits, Šuvakov-Dmitrašinović Catalogue, Origami ISA, Fano Plane, PG(2,2), Celestial Mechanics, Symmetry Reduction, G₂ Wall, Stability Boundary, Exceptional Algebras