This paper provides the formal mathematical proof that the discrete RGM attention lattice converges to classical smooth spacetime in the continuum limit, closing the remaining mathematical gaps in the quantum gravity derivation of Paper VI. Four results are established. (1) Convergence theorem: the discrete gravitational amplitude sum converges to the continuum integral with explicit error bound ε ≤ (1/28) · (LQF/r) ² at observation scale r. At nuclear scales the error is ~10⁻⁴⁰; at human scales ~10⁻⁷⁰. (2) Second-order metric theorem: the Gaussian attention weight profile produces only second-derivative metric variations in the continuum limit, making Lovelock's theorem applicable and the Einstein equations uniquely forced. (3) Newton's constant: GN = LQF²·c³/ℏ exactly, derived from the foam's attention weight normalization rather than dimensional analysis alone. (4) Lorentz invariance recovery: residual Lorentz violation is bounded by α· (p/MPl) ², consistent with the GW170817 bound |δv/c| < 10⁻¹⁵ by 29 orders of magnitude. The graviton dispersion relation E² = p²c² + α·p⁴·LQF²/ℏ² (Document 22) is identified as the leading lattice correction term. GW170817 provides the empirical anchor confirming the convergence rate at cosmological baselines. GRB 221009A (Ng and Steinbring 2025) independently confirms the holographic character of the foam established in Paper X. Together, Papers VI and XIII constitute a complete formal derivation of General Relativity from the RGM quantum foam. The convergence constant C = α is left as σ²/2 pending derivation of σ from first principles, which is addressed in Paper XIV. Builds on Papers I–XII of the RGM series.
Timothy Gleason (Thu,) studied this question.