Proof of Gₑff (N) = σ (N) ·GPDL 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 Gₑff (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) ·GPDL for any rank choice only because Gᵤnit 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 pₖ and pₗ are in disjoint variable spaces, so (A) ∧ (B) on pₖ constrains only its own ±1⁴ 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·Pₐctive (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 = Rₛurf/Rₜot is the gravitational engagement fraction of a single proton) and H2 (G ∝ engagement fraction, from the bridge formula D20) — Theorem 1 establishes Gₑff (N) = σ (N) ·GPDL for all N ≥ 0. The single remaining open problem is OP1: derive H1 from axioms C1–C4, i. e. , prove why Rₛurf = 310φ is the gravitational active surface of a single proton.