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

The Projective Dynamic Logo (PDL) framework predicts that the effective gravitational coupling takes the form G_eff(N) = σ(N)·G_PDL, where σ(N) = 1 − (1−κ)^N is the surface engagement fraction, κ = R_surf/R_tot = 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 H₀_local/H₀_CMB = √(σ(N_local)/σ(N_CMB)) = 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 d₂^sat 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: Δr_val(N) = σ(N)·Δr_val(∞) to better than 10⁻¹² for all N = 0…120, establishing G_eff(N) = σ(N)·G_PDL 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.