The Topological Big Bang and the Spontaneous Emergence of Metric Distance

In the current `N=64` complete-graph quench, the dimensionless singular state is finite and explicit at sweep `0`, with `2016` edges, `41664` triangles, and diameter `D=1`. Metric distance is born sharply at sweep `1`, where the graph remains connected but the diameter jumps to `D=2`, and the first macroscopic branch appears by sweep `16` with `D=4`, `275` edges, and only `82` triangles. The present result supports a finite topological phase transition from a hyper-connected pre-metric state to a sparse connected metric vacuum.