Emergent Einstein Gravity from the SO(3,3) Matrix Model on the Poincaré Homology Sphere

We assemble the complete derivation of four-dimensional Einstein gravity from the SO(3,3) matrix model compactified on the Poincare homology sphere P^3 = S^3/2I*, drawing on results from the companion paper series [1-8]. This closes the open calculation identified in [5, Section 17] as "the single most important open calculation" of the framework: the explicit demonstration that the saddle-point expansion around the fuzzy-P^3 configuration reproduces the 4D Einstein-Hilbert action with a computable Newton constant. The derivation chain is: (i) Matrix Hessian to Laplacian eigenvalues (2/3)k(k+2) on fuzzy P^3. (ii) Fuzzy Laplacian to smooth Laplacian, by the Thurston-Perelman uniqueness of the geometric structure on P^3. (iii) Scalar Laplacian to Dirac operator, by the Lichnerowicz identity D^2 = Delta + 1. (iv) Spectral data to topological invariants, by the Cheeger-Mueller theorem. (v) One-loop effective action to Einstein-Hilbert term, via Tr(H^{-1}) on P^3 and Steinacker's formula. (vi) Ghost freedom and graviton count, by Bromwich constraint. The effective Newton constant is G_4^{-1} = 42.86 R*^4/g^2, with the numerical coefficient determined by the regularised spectral zeta function zeta_total(1) = 3.572 = 60c, where c = 0.05953 is the universal Green's function constant on P^3. The 4D theory has exactly two graviton polarisations with positive kinetic energy, the Pati-Salam gauge group SU(4) x SU(2)_L x SU(2)_R, and three generations of chiral fermions. Together with the fine-structure constant alpha^{-1} = 137.035999177 and the vacuum energy Lambda = 0.183/R*^4, the framework determines all three gravitational observables -- G_4, Lambda, and the gauge coupling alpha -- as spectral-geometric invariants of the same manifold P^3. No free parameters enter. Numerical verification at N = 3, N = 5, and N = 7 confirms every structural prediction to machine precision: the vanishing action, the (N-1)^2 negative Hessian eigenvalues with exact SU(2) multiplicities, the exact Gaussianity of the antisymmetric sector (deviation < 10^{-14}), and the quadratic response of the one-loop determinant to spin-2 metric perturbations. A self-contained Python verification script (NumPy only, ~2 min runtime) is provided as supplementary material.