How does the magnetic flux density of a cylinder shaped magnet mathematically change as you reach the core? does it get stronger or weaker? by how much so, I DON'T mean core going from north to south, but from the outside edge of the diameter of the pole to the center of the diameter of the pole.
A magnet consists of magnetic dipoles. Hence, if you approach the dipoles (=tip of magnet), the magnetic field gets stronger. Once your "probe dipole" enters the magnet, I expect that the field would still increase. However, this should depend on the shape of the magnet.
At the University of Stuttgart Prof. Pfau does a similar things: They take atoms which posses a "strong" magnetic moment and cool them down. Eventually the atoms collapse due to the dipole-dipole interaction -- the atoms form molecules. If they change the shape of the atomic cloud, they observe changes in the collapse.