Black Holes in the Label Framework: Collapse, Streams, Vacuum Cores and the Label Count

The label framework describes matter as a field of comoving labels with volume-preserving relabellings. We collect its results on black holes. A homogeneous ball of label dust collapses as in the Oppenheimer–Snyder solution, and its labels reach the centre inside the horizon. Crossing dust shells create no label: each keeps its shell. Four-label vacua with different effective cosmological constants cannot be joined across a static spherical interface without a wall. Under horizon equilibrium and a premise on the entanglement entropy, the coefficient 1/4 fixes a lattice step a≈0.31√N Planck lengths for N minimally coupled massless scalars. The step counts no label states. For label matter quantised on embeddings, a classical argument shows that the labels of a hole formed from single-stream dust cannot leave it by radiation. Evaporation then ends in a remnant unless conservation of the unsigned label count fails. Microstates, unitarity and the singularity are not reached.