Emergent gravity from vacuum elasticity: exact geometric factor and first-principles derivation of Newton's constant

We derive classical gravity as the elastic response of a tensioned vacuum on T3 = R3 / L Z3, L = 50-75 Gpc IR regulator, with noncommutative substrate [x,y]=i theta, theta ~ Lp^2. The fundamental filament field T, |T|=1, is described by an O(3) sigma-model. Coarse-graining over V_coh = (l_coh_eff)^3, l_coh_eff = 0.1136 μm, yields isotropic distribution f = 1/(4pi) sin theta on S2 and exact orientation tensor Q = = 1/3 delta, Tr Q = 1. Flux of filaments through a plane is dN = rho_line |t·n| dA. Isotropic averaging gives <|t·n|> = <|cos theta|> = 1/2, Cauchy formula for line networks. For a filament tube radius r_S = 1 fm, chord through circular cross-section at impact parameter b is l(b) = 2 sqrt(r_S^2 - b^2). Physical distribution is uniform in b, p(b)db = db/r_S, giving mean chord = (1/r_S) integral_0^{r_S} 2 sqrt(r_S^2 - b^2) db = pi/2 * r_S. Area weighting 2pi b db/(pi r_S^2) would give 4/3 r_S for a 3D ball, not a 2D cylinder section. Combined, this yields geometric factor 1/8 = (pi/2)/(4pi) directly from Euler-Lagrange equations, no fitting. The Poisson equation gives G = chi_LQG * r_S^2 / (8 * kappa0_eff * (l_coh_eff)^2). With chi_LQG = 1.31e-22 from LQC bounce S_inst = 50.38, kappa0_eff = 0.18 GeV^2, r_S = 5.067 GeV^-1, l_coh_eff = 5.78e8 GeV^-1, we obtain G_pred = 6.99e-39 GeV^-2 vs observed G_N = 6.7088e-39 GeV^-2, 4.2 percent agreement with no free parameters. Exponential sensitivity Delta S=1 => Delta G/G = e.