10 Conclusion

We have proved four theorems, exact within the conformal metric toy model:

1. Ellipse theorem. ψ(t) = Ae^iωt is the unique minimum-complexity closed worldline. Keplerian orbits are selected by Solomonoff induction.
2. Curvature decomposition theorem. ρ = ρ_Ricci + ρ_tidal + ρ_graviton exactly. Tr(ρ_Weyl) = 0. Two polarisation modes emerge as degenerate eigenmodes.
3. Graviton amplitude theorem. ∥W(𝜖)∥ = sin(2𝜖)∕, identical in form to the Born rule identity of Paper VII. The conservation law ∥ρ_Ricci∥²+∥ρ_Weyl∥² = 1 mirrors ∥F∥²+∥B∥² = 1 of Paper VIII.
4. Newtonian potential theorem (proof sketch). Graviton flux conservation over 4πr² spherical shells gives A(r) ∝ 1∕r exactly, and V (R) = −GMm∕R with G = 1∕(8π) in Planck units. The factor 8π = 2 × 4π is the same spherical geometry factor that recovers Bekenstein–Hawking entropy in Paper IV.

Together with Papers VII and VIII, these results suggest that quantum mechanics and general relativity are two projections of the same compression principle onto different physical degrees of freedom. The conservation law ∥local∥² + ∥non-local∥² = 1 holds in both cases. The derivation of the Einstein field equations in the continuum limit is the primary outstanding goal.