Derivation of the Gravitational Constant: G = omega^2/(8*pi*rho)

Abstract

The gravitational constant G (6.674e-11 m^3 kg^-1 s^-2) is a measured constant in mainstream physics with no first-principles derivation. From energy-field axioms: matter = gaps (void structures) in the energy field; gaps are oscillating cavities (existence = fluctuation residence). The interaction of oscillating cavities is the secondary Bjerknes force (bubble dynamics, Crum 1975) - the gradient of a 1/r pressure field yields 1/r^2 attraction. Matching with Newton's law gives G = omega^2/(8*pi*rho), the only possible dimensional combination. The coefficient 8*pi = 4*pi (3D solid angle) x 2 (time average) is strictly derived; the n-dimensional generalization G_n = omega^2/(2*S_n*rho). Anchor omega = c/R_H = H0 gives rho = rho_crit/3 = 2.844e-27 kg/m^3, aligned with observed matter density (Omega_m = 0.31) within 7.5%, shared by five independent derivations (a0, black-hole four-formulae, acoustic horizon). Corollaries: event horizon = rigid-body binding boundary (R = 2GM/c^2 re-derivation); light = shear wave of the rigid body (mu = rho*c^2), gravitational waves at the same speed c (GW170817); gravity is weak because the rigid body is soft (c^4/G = 8*pi*K*R_H^2); equivalence principle from the same origin. Distinction from the Machian family (Sciama 1953; Uruena Palomo 2024).