Hyperbolic Measure Theory, Density Snaps, and the Exact Elimination of Cosmological and Black-Hole Singularities

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...