Paper CXCI: Nuclear Surface Density from Klein Quartic Hyperbolic Geometry: The Woods-Saxon Profile as Solid Angle in H³ We derive the nuclear surface density profile ρ(r) = ρ₀/(1 + exp((r−R)/a)) — the Woods-Saxon

We derive the nuclear surface density profile ρ(r) = ρ₀/(1 + exp((r−R)/a)) — the Woods-Saxon form — from the hyperbolic geometry of the Klein Quartic in 3+1 dimensional spacetime. The nuclear surface is modelled as a geodesic plane P₀ in H³. The solid angle subtended by P₀ from perpendicular depth t inside the nucleus is Ω(t) = 2π(1 − tanh(t/a)), an exact result of H³ geometry. The nuclear density at depth t is identified with the rate of change of this solid angle: g(t) = −dΩ/dt = (2π/a) sech²(t/a). Integration gives exactly the Woods-Saxon logistic profile. The surface diffuseness parameter a = arccosh(cosπ/3 / sinπ/7) × c/Λ_QCD = 0.5380 fm (at ℏ Λ_QCD = 200 MeV), in 2.7% agreement with the observed value 0.524 fm and exact at Λ_QCD = 205.3 MeV. The scale a is the inradius of the regular heptagonal tile in the {7,3} Klein Quartic tiling — the natural depth scale from the centre of each heptagonal domain to its boundary. This connects the nuclear surface to G₂ = Aut( ) through the chain G₂ PSL(2,7) = Aut(Fano) = 𝕆 ⊃ Aut(Klein Quartic). No free parameters are used beyond the QCD scale Λ_QCD. 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.