riskbert wrote:
Why does Pauli's principle hold? Is it a mathematical principle or a
heuristic argument? And why does the principle break in sufficiently
dense objects which then collapse to form neutron stars?

Because the electrons are arranged in rings.
When the ring rotates 360 degrees while precessing 720 degrees,
only electrons exactly opposite each other can follow
the same path.
http://users.accesscomm.ca/john/hydrogen.gif
http://rapfast.petcom.com/~john/2color2vieworbital.GIF
The rings 'stack' and grow into concentric rings
of 16 members (from inside as well as out, as is
evidenced by the elements such as Cobalt and Nickle
as well as the Lanthanides and Actinides- in red in the
GIF.)
http://rapfast.petcom.com/~john/periodicpattern.GIF
Not sure whqat happens under extreme conditions.
John