Does posterize black and white use same value ranges each time?

[Ofnuts]

Experimentally:

• Create an image with a grayscale gradient
  
• Make two duplicates, make one darker, one lighter, by moving the middle values in curves (so image sill contains pure black and pure white)
  
• Posterize all three to 8 levels
  

   

As you can see from the histograms of the results the levels are exactly the same in all three. Similar tests but make darker/lighter version by changing the darker/lighter values so that the results no longer contain pure white or pure black:

   

The Levels are the same in all three, even if some levels are not used in the picture...