Proof of G_eff(N) = σ(N)·G_PDL from the Trace Structure of K₄ and Independence of Gravitational Engagement

Gate 3 of the Projective Dynamic Logo (PDL) programme was established in D31 under a linear bridge inversion hypothesis. This paper replaces that hypothesis with an explicit proof chain under two named conditions, strengthening Gate 3 into a theorem with a precise and tractable residual. Four diagnostic blocks, verified exhaustively in Colab, establish the complete proof structure. Block 1 rules out two naive alternatives: σ(N)^18 fails by a factor 17.5 at N = 40, and the extensivity assumption G_eff(N) = N·G(1) fails with 450% error at N = 120. A tautology in the trace formula is identified and resolved: the formula gives σ(N)·G_PDL for any rank choice only because G_unit is defined to cancel the rank factor, so the true content lies elsewhere. Block 2 establishes two structural results: Proposition 1 proves that the cross-signs of distinct protons p_k and p_l are in disjoint variable spaces, so (A)∧(B) on p_k constrains only its own {±1}⁴ cross-sign space, yielding δ = P(joint) − P(marginal)² = 0 exactly (exhaustive over all 256 cross-sign pairs); and Lemma 2 proves that Tr(A) = 0 for the edge-adjacency matrix of K₄ (no self-adjacency), so for any S₄-invariant coupling Φ = αI₆ + βA, the term β vanishes from Tr(Φ·P_active(N)) for all β ∈ ℝ. Blocks 3 and 4 confirm that independence is contained in the definition of σ(N) as a union probability and is not an additional assumption. Under two named hypotheses — H1 (κ = R_surf/R_tot is the gravitational engagement fraction of a single proton) and H2 (G ∝ engagement fraction, from the bridge formula D20) — Theorem 1 establishes G_eff(N) = σ(N)·G_PDL for all N ≥ 0. The single remaining open problem is OP1: derive H1 from axioms C1–C4, i.e., prove why R_surf = 310φ is the gravitational active surface of a single proton.