The condition sface = 1/4 is equivalent, via the framework's normaliza- tion factors, to a constraint on the 7-state Shannon entropy: H7(a∗) = π p 3/8 = 1.9238 . . . This master relation can be veried symbolically: √ 2 π p 3/8/(2π √ 3) = √ 2 p 3/8/(2 √ 3) = p 3/4/(2 √ 3) = (1/2)/2 = 1/4 exactly. The constant π p 3/8 en- codes the complete geometric content of the black hole entropy: π from the Kruskal S1 factor, √ 3 from the 3-regularity of the Fano plane, 1/ √ 8 from the Kruskal two- sheet factor (1/ √ 2) and equatorial normalization (1/2). The self-consistent coupling is a∗= 0.53329128 . . ., determined uniquely by H7(a∗) = π p 3/8, with p∗ e7 = 0.22124 and p∗ other = 0.12979. The a2 correction from the Fano lattice dispersion is conrmed negligible (x2 = (ℓP /rT ∗ 2 )2 < 10−60 for all physical masses), so the exact 1/4 arises from the master relation alone. One new prediction is made (P44).Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.