Bekenstein–Hawking Entropy from the PDL Effective Gravitational Coupling: the 4π4π 4π Identity, Surface Counting, and Primordial Black Hole Predictions

The Gate 3 theorem of the Projective Dynamic Logo (PDL) framework establishes Geff(N)=σ(N)⋅GPDLGeff(N) = σ(N)· GPDL Geff(N)=σ(N)⋅GPDL with σ(N)=1−(1−κ)Nσ(N) = 1-(1-κ)^N σ(N)=1−(1−κ)N and κ=310φ/11017∈Q(5)κ = 310φ/11017 ∈ Q(√5) κ=310φ/11017∈Q(5) (D31/D36). Substituting this NN N-dependent coupling into the Bekenstein–Hawking formula SBH=kBc3A/(4Gℏ)SBH = k_B c^3 A / (4G) SBH=kBc3A/(4Gℏ), we derive algebraically that SBH/kB=4π(Meff/MPl)2SBH/k_B = 4π(Meff/MPl)^2 SBH/kB=4π(Meff/MPl)2 with Meff(N)=σ(N)⋅N⋅mpMeff(N) = σ(N)· N· m_p Meff(N)=σ(N)⋅N⋅mp, where MPl=ℏc/GPDLMPl = √{ c/GPDL} MPl=ℏc/GPDL is the PDL Planck mass. This identity, verified numerically to less than 10−1510⁻¹⁵ 10−15 relative error over ten decades of black hole mass, constitutes Proposition BH-2: the Bekenstein–Hawking entropy counts the square of the gravitationally active mass in Planck units. The universal constant C0=4π(mp/MPl)2=7.4221×10−38C_0 = 4π(m_p/MPl)^2 = 7.4221×10⁻³⁸ C0=4π(mp/MPl)2=7.4221×10−38 contains no free parameter; it is fully determined by the proton quintuplet (24,28,930,10087,11017)(24, 28, 930, 10087, 11017) (24,28,930,10087,11017) and the single QCD interface parameter Δmiso=md−muΔ mᵢₛₒ = m_d - m_u Δmiso=md−mu. Three parameter-free falsifiable predictions follow: (i) a 28.6%28.6% 28.6% suppression of black hole entropy at the CMB epoch (σ(40)2=0.7139σ(40)^2 = 0.7139 σ(40)2=0.7139); (ii) an upward shift of +11.89%+11.89% +11.89% in the primordial black hole survival threshold (M∗PDL=σ(40)−2/3⋅M∗GRM_*PDL = σ(40)-2/3· M_*GR M∗PDL=σ(40)−2/3⋅M∗GR), testable via Fermi-LAT gamma-ray constraints; (iii) a redshift-dependent entropy-to-energy ratio SBH/E∝σ(N(z))SBH/E ∝ σ(N(z)) SBH/E∝σ(N(z)) traceable in AGN populations. The ascending derivation — proving ln⁡Ωsurf=4π(Meff/MPl)2lnΩsurf = 4π(Meff/MPl)^2 lnΩsurf=4π(Meff/MPl)2 from the PDL axioms without invoking Schwarzschild geometry — is identified as Open Problem OP1 of D38 (Conjecture BH-3).