Density-Capped Interiors: The Radius at Which a Black Hole Reaches the Aldred Limit

For ten solar masses the horizon sits at 2.953e+04 metres while the capped radius is only 7.876e-10 metres, a ratio of 2.667e-14. The mean density inside that horizon is 1.843e+17 kilograms per cubic metre, below the bound by a factor 5.273e+40, so the horizon itself is nowhere near the ceiling: the cap is a deep-interior statement. For a billion solar masses the capped radius grows to 3.656e-07 metres but its ratio to the horizon falls to 1.238e-19, because the capped radius scales as the cube root of mass while the horizon scales linearly. The scaling is the result with the longest reach. Computed from stated assumptions, not measured.