We establish the Klein quartic M = PSL(2, 7)\H2 as the canonical brane surface in the One- Octonion Brane-Bulk Framework. The Klein quartic is the unique compact Hurwitz surface of genus 3 saturating the bound |Aut(M)| ≤84(g −1) with |Aut(M)| = 168 = |PSL(2, 7)|. We show: (i) PSL(2, 7) embeds into G2 as a finite subgroup; (ii) the three handles of the genus-3 surface correspond physically to the three Standard Model generations; (iii) the {7, 3} heptagonal tiling carries 24 heptagonal faces, 84 edges, 56 vertices, χ = −4; (iv) the Klein-quartic spine arccosh(3) = 1.7627 in warped metric corresponds to the η′-meson geodesic (Paper CCXL). Every framework geometric construction — brane modes, Klein-ladder eigenvalues, particle masses from winding numbers, mesic-nucleus geodesic lengths — factors through this single surface. We register five falsifiable structural predictions that distinguish framework physics from competing theories based on different brane topologies.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.