The PSL(2,7) Isomorphism, the Klein Quartic, and Proof That the Fano Lattice Re- quires Hyperbolic Embedding: Why the Universe Must Expand Paper LXIX of the Brane-Bulk Octonionic Series

The Fano plane has a discrete symmetry group of order 168: PSL(2, 7) = GL(3, 2) = Autproj(Fano). The identical group PSL(2, 7) is also the full automorphism group of the Klein Quartic a compact Riemann surface of genus 3 with constant negative curvature and in this capacity it is the unique Hurwitz surface at genus 3, achieving the maximum possible number of automorphisms 84(g−1) = 168 for a genus-3 surface (Hurwitz's theorem). This isomorphism is not a coincidence. It is a topological theorem that the 7-fold discrete symmetry of the Fano plane cannot be realised in at space: the Schwarz triangle group (2, 3, 7) that generates the Fano tiling has angle sum π/2 + π/3 + π/7 = 41π/42 < π, proving by the Gauss-Bonnet theorem that the tiling requires a surface of strictly negative curvature. The Fano plane, when continuously ex- tended to preserve all 168 of its PSL(2, 7) symmetries, must live on the Klein Quartic a hyperbolic Riemann surface of genus 3. The physical consequences for the brane-bulk framework are profound: (1) The AdS5 parent spacetime (Paper I) is not merely con- sistent with hyperbolic geometry it is algebraically required by the Fano lattice's own symmetry group. The negative curvature R ≈−20/ℓ2 P at the Planck scale (Paper LXII) is the curvature the (2, 3, 7) tiling demands. (2) The Schwarz triangle numbers (2, 3, 7) are precisely the Z2 isospin doublet, Z3 generation, and 7-node Fano structures of Pa- per LVI the same numbers appear in the Standard Model particle content because they encode the symmetry of the hyperbolic tiling. (3) The universe expands because the Fano lattice has an angular decit of π/42 per triangle in at space, and the brane expansion is the geometrical relaxation of this decit: as the AdS radius grows, the eective curvature K →0, and the angular decit is relieved. Cosmological expansion is not a dynamical accident but a topological inevitability of the (2, 3, 7) Schwarz triangle group. (4) The Klein Quartic's Euler characteristic χ = −4 and its tiling by 24 regular heptagons, 84 edges, and 56 vertices provide new numerical predictions for the brane- bulk framework. (5) The S7 (7-sphere) connects to the Klein Quartic through the Hopf 325 Brane-Bulk Octonionic Series B.D. Jagadeesan MD Chapter 4: Geometric Structure bration S7 →S4 and provides the compact bulk geometry whose fundamental domain is the Klein Quartic. Five new predictions follow (Predictions 141145). 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.