Quantitative Einstein Equation from G_eff(N) = σ(N)·G_PDL in the PDL Framework

The Projective Dynamic Logo (PDL) framework proves G_eff(N) = σ(N)·G_PDL as a theorem (Gate 3, D31/D36), where σ(N) = 1 − (1−κ)^N is the surface engagement fraction, κ = 310φ/11017 ∈ ℚ(√5), and N is the local coherent-closure density. This paper synthesises Gate 3 at the level of Einstein's field equations and makes the three epistemic layers of the PDL gravitational theory explicit. Definition 3.1 introduces the PDL gravitational coupling κ_PDL(N) = 8πσ(N)G_PDL/c⁴ = σ(N)·κ_Newton, which vanishes at N = 0, converges to the Newtonian κ as N → ∞, and is strictly increasing, with a deviation of 0.373% at N = 120. Proposition 4.2 derives the gravitational redshift formula ν_∞(r) = (1 − G_eff(N)M/(rc²))·ν₀, the standard general relativistic result with G replaced by G_eff(N), consistent with D33. Theorem 5.1 states the PDL Einstein equation G_μν[g_eff] = κ_PDL(N)·C_coh, where κ_PDL(N) is a theorem (Layer 1), C_coh is a structural target decomposed schematically as C_μν^(mass) + C_μν^(spin) + C_μν^(orbital) + C_μν^(leak) (Layer 2), and the PDL cosmological term Λ_PDL ~ C·η_L^18·(m_p c/ħ)² is a schematic estimate with C an open proportionality constant (Layer 3). The Hubble ratio prediction √(σ(120)/σ(40)) = 1.085868 is updated to 0.006% agreement with the observed ratio 73.04/67.26 = 1.085935 (SH0ES 2022), with zero free parameters, improving from 0.20% in D26 by direct use of the structural value N_CMB = 40 from D27.