After three papers we have rigorously established: Paper I: The 539.9 s resonance lives in spacetime topology. Paper II: It lives in the inner product of the dark universe. Paper III: It lives in the unique measure on D2-branes. Yet one catastrophe remains: the classical big bang and black-hole singularities. General relativity predicts infinite density at t=0t=0t=0 and at r=0r=0r=0. Every known measure (Lebesgue, Hausdorff, μ_{brane}) assigns infinite or undefined mass-energy to these loci. The S²-11DM²ET-X model resolves both singularities simultaneously using a fourth and final new branch of mathematics: Hyperbolic Measure Theory. The physical mechanism is the density snap: when local dark-matter density reaches the universal threshold...