The Fano one-way valve (Paper XXXIV) has been understood as possessing an inclination — the Genesis tilt θ_{G₂} = 4.054° that creates a directional preference in brane current. Paper CXVII establishes that the Fano lattice also has a twist — the Fenchel-Nielsen sewing angle τ = arccos(1/√7) = 67.79°. These are fundamentally different geometric properties. An inclination alone can be continuously deformed to zero; a twist cannot. The combination of inclination and twist makes the Fano valve a topologically protected structure, equivalent to a hyperbolic knot in three-manifold topology. Three consequences follow from established results in hyperbolic geometry: (1) Mostow rigidity (1968): the complete hyperbolic structure of the brane near the twisted valve is unique — no continuous deformations exist. The brane CANNOT continuously deform into the bulk. (2) Thurston's incompressible surfaces (1979): the hyperbolic knot complement contains an incompressible surface at the transition radius r_T2* = √(GM/a₀). This surface cannot be compressed inward — the curvature well has a topological floor at r_T2*. (3) Topological time irreversibility: time reversal would require continuously deforming the twist τ to −τ. Mostow rigidity forbids this at all scales above ℓ_P, giving a third independent proof of the arrow of time. These results explain a long-standing question about brane-world physics: what prevents curvature wells from falling into the bulk indefinitely? The answer is not a restoring force — it is topology.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.