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: $$D(x) = -\frac{\ln(x)}{\ln(\phi)}$$ and the identity $\phi^{D(x)} = 1/x$, the theory proves the energy invariant: $$E(\Psi) = 2\pi \quad \text{for all } \Psi > 0$$ Because the derivative of energy with respect to the field state is zero ($dE/d\Psi = 0$) everywhere, no coupling force can be derived from the potential. Consequently, each body evolves independently under the governing equation: $$\frac{d\Psi_i}{dt} = -\frac{1}{\Psi_i}(\Psi_i - x^*)$$ Where the attractor constant is $x^* \approx 0.710439$. 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 ($N$): $$\binom{27}{2} = 351 \le 369 < 378 = \binom{28}{2}$$ 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 $\binom{10}{3} = 120$ 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: $$r_{12} r_{13} r_{23} = \phi^{-214}$$ 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 $N=27$.