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=310φR_{\mathrm{surf}} = 310\varphi Rsurf=310φ cycles, φ\varphi 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)\wedge(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=11017R_{\mathrm{tot}} = 11017 Rtot=11017. This constitutes an unconditional structural derivation of the area law S∝RsurfS \propto R_{\mathrm{surf}} S∝Rsurf, which maps — via the Gate 3 theorem Geff(N)=σ(N)⋅GPDLG_{\mathrm{eff}}(N) = \sigma(N)\cdot G_{\mathrm{PDL}} Geff(N)=σ(N)⋅GPDL (D31/D36) — to the Bekenstein–Hawking scaling SBH∝AS_{\mathrm{BH}} \propto 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/Rtot\kappa = R_{\mathrm{surf}}/R_{\mathrm{tot}} κ=Rsurf/Rtot (OP1 of D36, partially resolved in D39).