Abstract

Paper VII established that bosons emerge as compression residuals of a purely fermionic system: the off-diagonal density matrix B = ρ − diag(ρ) encodes everything the codec records when a fermion moves, without any interaction term introduced by hand. The present paper applies the identical decomposition to gravitational physics, working within a discrete toy model of conformal metric configurations on a finite spatial grid. We prove four theorems, each exact within this model. Theorem 1: the minimum spectral-complexity closed worldline is ψ(t) = Ae^iωt — a single Fourier mode, hence an ellipse. Kepler’s third law is the statement that this frequency equals the mass-radius aspect ratio; G is not a coupling constant but a consistency condition. Theorem 2: the density matrix of any pair of conformal metric configurations decomposes exactly as ρ = ρ_Ricci + ρ_tidal + ρ_graviton, with Tr(ρ_Weyl) = 0 in every case and two degenerate graviton polarisation eigenmodes emerging automatically. Theorem 3: the graviton amplitude follows ∥W(𝜖)∥ = sin(2𝜖)∕, the exact GR analogue of the Born rule identity ∥B(𝜃)∥ = sin(2𝜃)∕ from Paper VII. Theorem 4: graviton flux conservation over spherical shells of area 4πr² gives field amplitude A(r) ∝ 1∕r exactly and Newtonian potential V (R) = −GMm∕R with G = 1∕(8π) in Planck units, where the factor 8π = 2 × 4π arises entirely from spherical geometry — the same factor that recovers Bekenstein–Hawking entropy in Paper IV. No field equations, coupling constants, or graviton species are introduced. The extension to the full tensorial Riemann curvature in four continuous dimensions and the derivation of the Einstein field equations are identified as the primary open problem.