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May 24, 2026Open Access

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

Key Points

The aim is to derive four-dimensional Einstein gravity from the SO(3,3) matrix model on the Poincaré homology sphere.
Derivation of Laplacian eigenvalues on fuzzy P^3 using matrix Hessian,
Transition from fuzzy to smooth Laplacian through Thurston-Perelman's geometric uniqueness,
Conversion of scalar Laplacian to Dirac operator via the Lichnerowicz identity.
Effective Newton constant determined as G_4^{-1} = 42.86 R*^4/g^2 with computed numerical coefficients.
The model predicts all three gravitational observables — G_4, Lambda, and gauge coupling alpha — as spectral-geometric invariants.
Numerical validation confirms structural predictions with actions and eigenvalues to machine precision.

Abstract

We assemble the complete derivation of four-dimensional Einstein gravity from the SO (3, 3) matrix model compactified on the Poincare homology sphere P³ = S³/2I*, 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-P³ 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 P³. (ii) Fuzzy Laplacian to smooth Laplacian, by the Thurston-Perelman uniqueness of the geometric structure on P³. (iii) Scalar Laplacian to Dirac operator, by the Lichnerowicz identity D² = 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 P³ and Steinacker's formula. (vi) Ghost freedom and graviton count, by Bromwich constraint. The effective Newton constant is G₄^-1 = 42. 86 R*⁴/g², with the numerical coefficient determined by the regularised spectral zeta function zetaₜotal (1) = 3. 572 = 60c, where c = 0. 05953 is the universal Green's function constant on P³. 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 -- G₄, Lambda, and the gauge coupling alpha -- as spectral-geometric invariants of the same manifold P³. 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.

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

Key Points

Abstract

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