Surface Locality of Relational Entropy and the PDL Area Law

Within the Projective Dynamic Logo (PDL) framework, every composite structure is organised as a relational closure whose interior communicates with the exterior exclusively through a finite active surface (Rsurf=310φRsurf = 310φ Rsurf=310φ cycles, φφ φ the golden ratio). Paper D29 of the corpus proved that the valence sector of the PDL proton occupies a unique configuration satisfying the (A)∧(B)(A)(B) (A)∧(B) stability criterion; its accessible microstate count is therefore exactly one. The present paper formalises this observation as Proposition BH-1: the information-theoretic entropy of a PDL closure is carried entirely by the surface sector, not by the total relational budget Rtot=11017Rₜₒₜ = 11017 Rtot=11017. This constitutes an unconditional structural derivation of the area law S∝RsurfS ∝ Rsurf S∝Rsurf, which maps — via the Gate 3 theorem Geff(N)=σ(N)⋅GPDLGeff(N) = σ(N)· GPDL Geff(N)=σ(N)⋅GPDL (D31/D36) — to the Bekenstein–Hawking scaling SBH∝ASBH ∝ A SBH∝A at macroscopic scale. The result is unconditional: it follows from the four PDL axioms together with the uniqueness theorem of D29, without invoking the Schwarzschild geometry or any thermodynamic postulate. The coefficient 1/41/4 1/4 in the Bekenstein–Hawking formula is identified as a quantitative open problem conditional on the algebraic proof of κ=Rsurf/Rtotκ = Rsurf/Rₜₒₜ κ=Rsurf/Rtot (OP1 of D36, partially resolved in D39).