Why does spacetime curve in proportion to energy-momentum? Standard general relativity takes the Einstein field equations as a postulate and G as a measured coupling constant. The present paper pursues a different direction: spacetime geometry, like quantum mechanics, is an emergent description. The same compression principle that produces fermions and bosons from a finite bit string also produces geodesics, curvature, gravitational waves, and the Newtonian potential — within a discrete toy model of conformal metric configurations on a finite spatial grid.
The central claim is that the density matrix decomposition
applied to metric configurations instead of fermionic configurations gives the Ricci/Weyl split of the Riemann tensor in this toy model. The diagonal part is local — matter here curves space here. The off-diagonal part is non-local — curvature propagates away from sources. The first is the Einstein equation source term. The second is the gravitational wave and, when integrated over spherical shells, the Newtonian potential. Neither is postulated; both follow from the definition B = ρ − diag(ρ) and the geometry of three-dimensional space.
Remark (Scope of the toy model). All theorems in this paper are proved for conformal metric configurations g(x) ∈ ℝ on a discrete spatial grid of L sites, encoded as ψ = g(0) + ig(1). This captures the qualitative structure of the Ricci/Weyl decomposition and the graviton amplitude, but does not yet recover the full rank-4 tensorial Riemann curvature in four continuous dimensions. The extension to the Einstein field equations in the continuum limit is the primary open problem; see Section 8.
The identical formula sin(2𝜃)∕
governs both the photon amplitude (Paper VII) and the graviton
amplitude (Theorem 3 below). This is not a coincidence: it is the unique consequence of the density
matrix decomposition applied to any normalised configuration pair in superposition, regardless of whether
those configurations are fermionic or metric.
This paper is structured as follows. Section 2 proves the ellipse theorem (Theorem 1). Section 3 proves the Ricci/Weyl decomposition (Theorem 2). Section 4 proves the graviton amplitude theorem (Theorem 3) and derives the metric-configuration conservation law. Section 5 derives the two polarisation modes. Section 6 proves the Newtonian potential theorem (Theorem 4). Section 7 places G as a consistency condition. Section 8 states the open problems. Section 9 discusses the results.