The central result across Papers VII and VIII is that a single operation,
applied to compressed configurations of any physical degree of freedom, produces the local/non-local split underlying both quantum field theory and general relativity.
For fermionic configurations (Paper VII, Paper VIII): the diagonal gives Pauli exclusion; the off-diagonal gives the photon, gluon, and Born rule.
For conformal metric configurations (this paper): the diagonal gives the Ricci source term; the off-diagonal gives the Weyl tensor, gravitational waves, and — integrated over spherical shells — the Newtonian potential.
The shared formula sin(2𝜃)∕
for both boson amplitude (Paper VII) and graviton amplitude
(Theorem 3) is the signature of this unity: the formula is the unique consequence of the density matrix
decomposition applied to any normalised configuration pair, regardless of the physical degrees of freedom
involved.
The conservation law ∥ρRicci∥2 + ∥ρWeyl∥2 = 1 (Corollary 3) is the metric-configuration analogue of ∥F∥2 + ∥B∥2 = 1 (Paper VIII, Theorem 3). Both are instances of the same Pythagorean identity on Hilbert space, confirming that quantum mechanics and general relativity share not only a formula but a conservation structure.
The present results are exact within the toy model. The extension to the full tensorial Riemann curvature in four continuous dimensions and the derivation of the Einstein field equations require a treatment of the continuum limit that is beyond the scope of this paper (Open Problem 2 and 3).
The proportionality ∥W∥∝M assumed in Theorem 4 is physically natural but has not been derived from the counting equation alone (Open Problem 1).
Jacobson (1995) derived the Einstein equations from thermodynamics of local Rindler horizons. Verlinde (2011) derived Newtonian gravity from entropic forces. The present approach differs: it derives geodesics, curvature decomposition, graviton polarisations, and the Newtonian potential from spectral complexity minimisation alone, without postulating thermodynamics, entropy, or holography. These emerge as consequences, not inputs.