Non-Orthogonal Planes in the 7D Octonionic Parent, the Projective Special Linear Group PSL(2,7), and the Unied Geometric Origin of Hyperbolic Bulk Geometry, Logarithmic Coupling Running, and the Yukawa Hierarchy

Previous papers in this series treated the hyperbolic geometry of the AdS5 bulk as a consequence of the Schwarz triangle group (2, 3, 7) requiring a negatively curved embedding (Paper LXIX), and treated the logarithmic running of coupling constants as following from this hyperbolic geometry (Paper LXII). We now identify a more funda- mental and unied origin for both: the non-orthogonality of the seven imaginary planes e1,. . . , e7 in the 7D octonionic parent spacetime. In the standard octonionic metric, these seven planes are mutually orthogonal (⟨ei, ej⟩= −δij), the geometry is at, and the symmetry group is Z7 2. Once we allow non-orthogonality (⟨ei, ej⟩= −δij +γ, γ ̸= 0), three consequences are automatic, with no additional assumptions: (1) Hyperbolic geometry is forced. The Gram matrix G = − (1 + γ) I + γJ has sectional curvature K = −γ2/ (1 −γ2) 0. The AdS5 bulk geometry is not inserted it is the natural metric for any non-zero non-orthogonality. (2) PSL (2, 7) is the automor- phism group. The automorphism group of the non-orthogonal 7-plane conguration preserving the Gram matrix is the projective special linear group PSL (2, 7) ∼= GL (3, 2) of order 168 the same group that is the automorphism group of the Fano plane and the Klein Quartic (Paper LXIX). It is not chosen; it is forced by the geometry of the non-orthogonal planes. (3) Logarithmic coupling running and the Yukawa hier- archy share a single parameter γ. The non-orthogonality γ controls the curvature of the hyperbolic bulk, hence the logarithmic rate of coupling running; and controls the projection of each generation plane onto the condensate direction, hence the Yukawa coupling of each fermion generation. The fermion mass hierarchy yt ≫yb ∼yc ≫ye follows from the Z3 generation symmetry of H+ = e0, e1, e2, e3: the three generations are the three Z3 steps in H+, and y (n) ∼γ3−n eff where n is the generation index and γeff is the eective non-orthogonality at the fermion mass scale. Furthermore, QCD con- nement has a geometric interpretation: it occurs when γeff→1/6, the value at which the Gram matrix G acquires a zero eigenvalue and the colour-sector metric degenerates. Four new predictions follow (Predictions 168171). 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.