Deep-Future Sump Depletion and the {7, 3} ↔{3, 6} Tiling Interface: The Geometric Origin of the 3:2 QPO Ratio

In the One-Octonion Brane-Bulk Framework, two regimes coexist: the hyperbolic {7, 3} tiling of the Klein quartic (genus-3, three generations, the brane's present state), and the Euclidean {3, 6} A2-hexagonal tiling (the asymptotic future when sump curvature vanishes globally). Compact-object physics samples the boundary interface between these two tilings: {7, 3} inside the sump-dominant zone (within rT∗ 2 ), {3, 6} outside it. The 3:2 universal QPO ratio (Paper CXIV) emerges from this interface as the ratio of the two tilings' valencies: "3" from {7, 3} (three heptagons at each vertex), "2" from {3, 6} projected onto the brane time direction (Klein-ladder rung k = 2, where the pion lives). This paper makes the geometric identification explicit, derives the deep-future evolution κeff(t) →0 as t →Tsump (∼72.65 Gyr), and registers five falsifiable predictions including the 3:2 ratio's dependence on cosmological epoch (the ratio drifts from 3:2 at low z toward 1:1 as the universe ages, becoming exact-1:1 only at t = Tsump). 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.