Bekenstein–Hawking Entropy from the PDL Effective Gravitational Coupling: the 4π4\pi 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)⋅GPDLG_{\mathrm{eff}}(N) = \sigma(N)\cdot G_{\mathrm{PDL}} Geff(N)=σ(N)⋅GPDL with σ(N)=1−(1−κ)N\sigma(N) = 1-(1-\kappa)^N σ(N)=1−(1−κ)N and κ=310φ/11017∈Q(5)\kappa = 310\varphi/11017 \in \mathbb{Q}(\sqrt{5}) κ=310φ/11017∈Q(5) (D31/D36). Substituting this NN N-dependent coupling into the Bekenstein–Hawking formula SBH=kBc3A/(4Gℏ)S_{\mathrm{BH}} = k_B c^3 A / (4G\hbar) SBH=kBc3A/(4Gℏ), we derive algebraically that SBH/kB=4π(Meff/MPl)2S_{\mathrm{BH}}/k_B = 4\pi(M_{\mathrm{eff}}/M_{\mathrm{Pl}})^2 SBH/kB=4π(Meff/MPl)2 with Meff(N)=σ(N)⋅N⋅mpM_{\mathrm{eff}}(N) = \sigma(N)\cdot N\cdot m_p Meff(N)=σ(N)⋅N⋅mp, where MPl=ℏc/GPDLM_{\mathrm{Pl}} = \sqrt{\hbar c/G_{\mathrm{PDL}}} 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 = 4\pi(m_p/M_{\mathrm{Pl}})^2 = 7.4221\times10^{-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−mu\Delta m_{\mathrm{iso}} = 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=0.7139\sigma(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_*^{\mathrm{PDL}} = \sigma(40)^{-2/3}\cdot M_*^{\mathrm{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))S_{\mathrm{BH}}/E \propto \sigma(N(z)) SBH/E∝σ(N(z)) traceable in AGN populations. The ascending derivation — proving ln⁡Ωsurf=4π(Meff/MPl)2\ln\Omega_{\mathrm{surf}} = 4\pi(M_{\mathrm{eff}}/M_{\mathrm{Pl}})^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).