Derived! Planck--Einstein Vacuum using Charlton Geometry

We discharge the outstanding Einstein–Hilbert obligations in Charlton Geometry (CG) by proving the unique long-scale, second-derivative, diffeomorphism-invariant metric closure induced by the ω-carpet on the E4 weave. Under the CG macro-Lift, the induced metric g = η(ω) admits an effective field theory (EFT) expansion controlled by the minimum link/core scale rmin. In four dimensions, Lovelock uniqueness implies that the leading local term yielding second-order metric equations is Einstein–Hilbert plus a cosmological term, with higher-curvature corrections suppressed by rmin. Vacuum (regulator-free) regions therefore satisfy Gμν + Λgμν = 0 (and Rμν = 0 when Λ = 0). Finally, importing the CG→CO interface that exports ℏ, we give the Planck bridge: ℓ2P = ℏG/c3 ∼ ζ r2 min with a single dimensionless renormalization constant ζ.