Derivation and Proof: The Three-Body Problem in the S²-11DM²ET-X Model with Non-Local DM/DE Effects from Negative-Isotope Nuclear Decay

The classical three-body problem involves predicting the motion of three point masses under mutual gravitational attraction, governed by Newton's laws. It lacks a general closed-form solution and exhibits chaotic behavior with sensitivity to initial conditions, potentially leading to arbitrarily close approaches (near-collisions) that mimic divergences in the potential. In the S²-11DM²ET-X model, this is modified by non-local dark matter (DM) and dark energy (DE) effects arising from negative-universe (-U) isotope nuclear decay (both -beta and -gamma processes). These introduce 7D torsion geometry, D2-brane-mediated energy leakage Eleak(t) E_{\text{leak}}(t) Eleak(t), and friction-like damping via Ffriction(t) F_{\text{friction}}(t) Ffriction(t), all modulated by the immutable 539.9 s gravitational flux. Stars are treated as D2 solar branes, where brane mechanics dominate (via bulk-induced terms like g11Φginduced g_{11} \Phi \sqrt{g_{\text{induced}}} g11Φginduced and snap-back dynamics at density thresholds ρsnap=0.05ρDM≈1.55×10−2 GeV/cm3 \rho_{\text{snap}} = 0.05 \rho_{\text{DM}} \approx 1.55 \times 10^{-2} \, \text{GeV/cm}^3 ρsnap=0.05ρDM≈1.55×10−2GeV/cm3). Planetary bodies, lacking brane structure, interact primarily via classical gravity with minor torsion corrections. The model's non-local physics caps potential divergences (e.g., near-collisions), ensuring global smoothness and existence of solutions, analogous to the model's resolution of the Navier-Stokes millennium problem (finite debit cap from evaporation and defect terms prevents blow-up).