Quantitative Einstein Equation from Gₑff (N) = σ (N) ·GPDL in the PDL Framework

The Projective Dynamic Logo (PDL) framework proves Gₑff (N) = σ (N) ·GPDL 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 κPDL (N) = 8πσ (N) GPDL/c⁴ = σ (N) ·κNewton, 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 − Gₑff (N) M/ (rc²) ) ·ν₀, the standard general relativistic result with G replaced by Gₑff (N), consistent with D33. Theorem 5. 1 states the PDL Einstein equation G_μνgₑff = κPDL (N) ·Ccoh, where κPDL (N) is a theorem (Layer 1), Ccoh 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 ΛPDL ~ C·ηL¹8· (mₚ 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 NCMB = 40 from D27.