Theorem 4: The Newtonian Potential from Graviton Flux

The graviton amplitude theorem gives ∥W∥_max = 1∕ √ --
2

for a single metric hop at 𝜖 = π∕4. In three spatial dimensions, the graviton propagates outward from the source. By conservation of graviton flux over spherical shells, the field amplitude at distance r satisfies

Theorem 4 (Newtonian potential from graviton flux). Let source mass M produce graviton field amplitude A_M(r) = ∥W_M∥∕(2 --
√ π r) and test mass m produce A_m(r) = ∥W_m∥∕(2r), both evaluated at distance r from their respective centres. With ∥W∥∝M (more bits produce stronger field; see Open Problem 1 in Section 8), the interaction potential from the field overlap integral

∫ ∞ V (R ) = − AM (r)Am (r)4πr2 dr, ℓP

evaluated with lower cutoff r_min = ℓ_P (Planck length), gives

|--------------------------------------------| |V (R ) = − GM-m-, G = -1- (Planck units). ------------R-------------8π-----------------|

The factor 8π = 2 × 4π arises entirely from spherical geometry.

Proof sketch. Substituting A_M(r) = ∥W_M∥∕(2 √ π- r) and A_m(r) = ∥W_m∥∕(2 √ π- r) into the overlap integral (both amplitudes evaluated at the same radial distance r from the interaction midpoint, which introduces the factor 1∕R from the separation):

∥WM ∥ ⋅∥Wm ∥ ∫ ∞ 1 ∥WM ∥ ⋅∥Wm ∥ V(R ) = −------------- -2 dr ⋅R = − -------------. 4π ℓP r 4πR

With ∥W_M∥ = M∕ √--
2 and ∥W_m∥ = m∕ √ --
2 :

M-m-- V (R) = − 8πR ,

giving G = 1∕(8π) in Planck units.

Note: this is a proof sketch, not a full derivation. The proportionality ∥W∥ ∝ M is assumed; see Open Problem 1. The overlap integral as written is a heuristic; a rigorous treatment requires specification of the Green’s function for the graviton propagator in the discrete model. □

The 4π factor and consistency with Paper IV

The factor 4π appearing in the spherical shell area is the same factor that appeared in Paper IV when recovering the Bekenstein–Hawking entropy: flat raster geometry must be corrected to spherical geometry by multiplying by 4π. Its appearance here confirms geometric consistency: 4π is not a free parameter but the unique conversion between discrete counting and spherical propagation, appearing wherever the model transitions from flat to spherical three-dimensional geometry.

Numerical verification

The amplitude product A(r) ⋅r = ∥W∥_max∕(2 √ π-

) is constant to machine precision across all r, confirming exact 1∕r falloff. The force F(r) = −dV∕dR ∝ 1∕R² follows analytically from V ∝ 1∕R. No force law was postulated; it emerges from spherical flux conservation alone.

6 Theorem 4: The Newtonian Potential from Graviton Flux

The 4π factor and consistency with Paper IV

Numerical verification