Two-Level Crossover Resolution,

The brane-bulk octonionic framework predicts that the sump (dark matter analogue) carries a pressure perturbation set by a two-level Fano crossover. Level 1 describes the background Fano node coherence: c2⋆ s (η) = (1/7)/(1 + H/H⋆), where H⋆≃10,634 km/s/Mpc is the geometric mean √H0Hrec, ensuring the sump is pressure- less at recombination (H ≫H⋆) while supporting a late-time pressure c2 s →1/7 today. Level 2 is the mode-dependent Maxwell sump crossover: c2 s,sump(k, η) = c2⋆ s (η)/(1+Win), where Wi = Hphys p a/(2k a0) is the Weissenberg number. At CMB scales Wi ≫1, sump clusters exactly as cold dark matter. At sub-a0 scales Wi ≪1, sump becomes vis- cous and generates the MOND acceleration law. We implement this two-level crossover in CLASS via a three-parameter extension and demonstrate numerically: (1) TT resid- uals vanish to machine precision at all ℓwhen H ≫H⋆; (2) S8 tension is resolved at 0.03σ via the lensing kernel mechanism S8,lens = S8,CMB× p 6/7 = 0.7697 (observed: 0.769 ± 0.024), with zero free parameters; (3) The c(t) ∝H(t) self-protection the- orem guarantees that all conformal-time integrals R cs(η) dη are identical to ΛCDM, protecting peak positions at all multipoles. The lensing kernel mechanism is physical: c2⋆ s (z = 0) = 1/7 means one-seventh of the sump is pressure-supported and invisible to gravitational lensing, giving Ωm,eff= (6/7)Ωm for all lensing observables while leaving the CMB TT power unchanged. One prediction (P16) is conrmed. The CLASS im- plementation provides the foundation for future derivations of ns and As from brane ination. 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.