Paper CLXXXV: Fano Node Self-Coupling at the Gudermannian Saturation and the Universal Three-Class Structure of Brane-Bulk Transit Fractions: pp → HH → bbγγ Energy Cascade and κλ = 1

We show that the pp → HH → bb̄γγ process, interpreted within the one-octonion brane-bulk framework, reveals three universal classes of brane-bulk transit fraction f_bulk = 1 − sinc²(x), each determined by exact kinematics with zero free parameters. Class I (inter-rung hop, r ≈75%): x = (m_A² − m_B²)/(4m_A²) ≈ 0.107–0.119, f ≈ 3.7–4.6%. The top-Higgs hop (x = 0.1187, f = 4.55%) is the direct analog of the pion-muon hop in the hadronic vacuum polarization (x = 0.1067, f = 3.69%), both having mass ratio r ≈ 0.73–0.76. Class II (massive particle → massless photon): x = 1/4 exactly for any emitting mass, giving f_bulk = 1 − sinc²(1/4) = 18.943% universally. This applies to W and top loops in H→γγ (identically), and approximately to H→bb̄ (x ≈ 1/4 since m_b m_H). Class III (self-coupling): x = 1/2 exactly for any particle coupling to itself, giving f_bulk = 1 ≪ − (2/π)² = 59.471% — the Gudermannian saturation derived in Paper CLXXXIII. The HHH trilinear vertex sits at this saturation point, giving κλ = 1.000 (SM) to leading order. All 22 available Higgs coupling observables from ATLAS and CMS are consistent with the framework at mean |σ| = 0.25σ. Three falsifiable predictions are given for HL-LHC. 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.