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 fbulk = 1 − sinc² (x), each determined by exact kinematics with zero free parameters. Class I (inter-rung hop, r ≈75%): x = (mA² − mB²) / (4mA²) ≈ 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 fbulk = 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 mb mH). Class III (self-coupling): x = 1/2 exactly for any particle coupling to itself, giving fbulk = 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.