Bekenstein–Hawking Entropy from the PDL Effective Gravitational Coupling: the 4π4 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) ⋅GPDLG₄₅₅ (N) = (N) G₏₃₋ Geff (N) =σ (N) ⋅GPDL with σ (N) =1− (1−κ) N (N) = 1- (1-) N σ (N) =1− (1−κ) N and κ=310φ/11017∈Q (5) = 310/11017 Q (5) κ=310φ/11017∈Q (5) (D31/D36). Substituting this NN N-dependent coupling into the Bekenstein–Hawking formula SBH=kBc3A/ (4Gℏ) S₁₇ = kB c³ A / (4G) SBH=kBc3A/ (4Gℏ), we derive algebraically that SBH/kB=4π (Meff/MPl) 2S₁₇/kB = 4 (M₄₅₅/M₏₋) ² SBH/kB=4π (Meff/MPl) 2 with Meff (N) =σ (N) ⋅N⋅mpM₄₅₅ (N) = (N) N mₚ Meff (N) =σ (N) ⋅N⋅mp, where MPl=ℏc/GPDLM₏₋ = c/G₏₃₋ 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. 4221×10−38C₀ = 4 (mₚ/M₏₋) ² = 7. 422110^-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 m₈ₒ₎ = md - mᵤ 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 (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 = (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) ) S₁₇/E (N (z) ) SBH/E∝σ (N (z) ) traceable in AGN populations. The ascending derivation — proving ln⁡Ωsurf=4π (Meff/MPl) 2ₒₔₑ₅ = 4 (M₄₅₅/M₏₋) 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).