The N-Body Problem on the Brahim Manifold

The N-Body Problem on the Brahim Manifold extends the Brahim Field Equations from single-body dynamics to multi-body configurations on a shared manifold. 1. Field Foundation (Recap) Using the dimension map: D(x) = -ln(x)/ln(φ) and the identity φD(x) = 1/x, the theory proves the energy invariant: E(Ψ) = 2π for all Ψ > 0 Because the derivative of energy with respect to the field state is zero (dE/dΨ = 0) everywhere, no coupling force can be derived from the potential. Consequently, each body evolves independently under the governing equation: dΨᵢ/dt = -1/Ψᵢ(Ψᵢ - x^*) Where the attractor constant is x^* ≈ 0.710439. 2. Constraint Geometry Instead of force integration, multi-body structure is introduced as constraint geometry. Allowed pair-distance scales are drawn from the ten Brahim numbers: B = \27, 42, 60, 75, 97, 117, 139, 154, 172, 187\ and their non-empty subset sums. Exhaustive enumeration of these combinations yields 369 distinct admissible scales, organized into a level hierarchy. The N=27 Boundary Requiring distinct pair scales provides an algebraic boundary for the maximum number of bodies (N): 272 = 351 ≤ 369 < 378 = 282 Thus, the Brahim manifold admits configurations up to $N=27$ under the distinctness constraint. 3. Mirror Symmetry and the Generating Triangle The theory defines mirror symmetry via the involution: $$M(x) = 214 - x$$ Among the 103 = 120 candidate triples, there exists a unique generating triangle: \42, 75, 97\ The pairwise sums of this triangle reproduce mirrored Brahim numbers. This geometric property yields the conservation law: r₁₂ r₁₃ r₂₃ = φ⁻²¹⁴ This law has been verified to high precision. 4. Reproducibility The framework includes a worked three-body example and a reproducible Python appendix (utilizing numpy and itertools) to implement proofs and validation tests. The final scope of the work lists open problems regarding: Physical interpretation. Modular gaps. Extensions beyond $N=27$.