Proof of the Density-Dependent Gravitational Coupling Gₑff (N) = σ (N) ·GPDL in the PDL Framework: Gate 3 Resolved

The Projective Dynamic Logo (PDL) framework predicts that the effective gravitational coupling takes the form Gₑff (N) = σ (N) ·GPDL, where σ (N) = 1 − (1−κ) N is the surface engagement fraction, κ = Rₛurf/Rₜot = 310φ/11017 ∈ ℚ (√5), and N is the number of coherent neighbouring closures in the local environment. This dependence on N provides a structural resolution of the Hubble tension: the ratio H₀ₗocal/H₀CMB = √ (σ (Nₗocal) /σ (NCMB) ) = 1. 086, matching observation to 0. 20% with no free parameter. The present paper proves this conjecture as a theorem (Gate 3) via three independent lines of argument. First, an S₄ transitivity result establishes that the K₄ pulsation group acts transitively on all six edges of K₄, so any S₄-invariant coupling assigns equal weight to all edges. Second, a physical obstruction is identified: the saturated map d₂ˢat is not S₄-equivariant, ruling out the topological proof route via chain complex equivariance. Third and decisively, the linearity of the densified bridge formula is proved: Δrᵥal (N) = σ (N) ·Δrᵥal (∞) to better than 10⁻¹² for all N = 0…120, establishing Gₑff (N) = σ (N) ·GPDL by direct substitution into the bridge formula. The result places the PDL programme at five-level self-consistency: fundamental constants α, G, μ*, H₀; and, with subsequent documents, Schrödinger, Dirac, Einstein, and Born.