Derivation of the Bekenstein–Hawking One-Quarter Coefficient from the PDL Axioms: Resolution of OP12/BH-3

The coefficient of one-quarter in the Bekenstein–Hawking entropy formula SBH = kB c³A/ (4Gℏ) has long lacked a combinatorial derivation within the PDL programme (OP12/BH-3 in DM v18). This document derives it as an unconditional theorem of the four PDL axioms C1–C4. The derivation proceeds in three steps. First, Lemma 1 establishes by exhaustive enumeration over all 768 configurations of D29 that exactly 4 cross-sign configurations out of 16 satisfy the (A) ∧ (B) stability criterion for any given surface relation: the stable fraction per relation is precisely 1/4. Second, Theorem 1 proves statistical independence of the surface couplings using the three lemmas of D42 — equiparticipation (D42-L1), cross-triangle blindness (D42-L2), and S₄-equivariance verified over 24, 576 cases with zero violation (D42-L3) — together with the Indifference Lemma H3, itself an unconditional theorem of C1–C4. Under independence, the number of accessible stable configurations on Rₛurf surface relations is Ωₛurf = 4Rₛurf, where Rₛurf = 310φ ∈ ℚ (√5) is the PDL proton active surface. Third, Corollary 1 identifies the resulting surface entropy Sₛurf = kB · Rₛurf · ln 4 with the Bekenstein–Hawking formula: the one-quarter coefficient is the stable fraction 4/16 = 1/4, and each surface relation contributes exactly log₂ 4 = 2 bits of entropy. The derivation is entirely combinatorial. No semiclassical gravity, no Barbero–Immirzi parameter, and no string-theoretic microstate counting are invoked. The same fraction 1/4 that appears in the Bekenstein–Hawking formula also underlies the London equation derived in D49, reflecting the single combinatorial engine of the PDL axioms. This document resolves OP12/BH-3 of DM v18.