Well, I have edited my edit to remove what was offensive. I'm sorry, I should have started like that.

All I am saying is that something has to cool for the degenerate core to form. Even if it already exists in the carbon core of a large star, it did not exist during the hydrogen-burning period, and most of the helium burning either. Once degenerate it does not give up any heat, but heat must get out, somehow, for it to form in the first place. If the matter of the entire star was not hot, any star or more than 1.5 solar mass would collapse in a neutron star or a black hole.

Apparently I cannot change my vote on your post unless yo uedit it. I believe an extra blank at the end of any paragraph would do.

Let's both cool down, your answer has got plenty of upvotes. Yes, the material that becomes the AGB core and then the white dwarf must cool, but it becomes hotter - a natural consequence of the virial theorem. The reason the core becomes degenerate is not because its temperature gets lower it is because it becomes more dense.

The white dwarf forms by shedding its envelope and it moves to the top left of the HR diagram. From there is cools at almost constant radius (and constant Fermi energy). The value of the Fermi energy is in the range 1-10 MeV, depending on the mass of the white dwarf ("of order MeV). Electrons have energies from zero up to the Fermi energy.

Have aplay with this widget I made (a long time ago, not for this discussion!). You can set the T to 1E8K and the density to 10^10 kg/m^3 (about right for the centre of an average white dwarf. You can see the Fermi energy is about 1 MeV and the degeneracy parameter is 60 before it has even started on the WD cooling phase.

geogebra.org/m/ACZt5h7a

In terms of heat getting out. Of course the ions are basically an ideal gas (until they solidify). It is they that contain all the thermal energy and it is they that cool, not the electrons.

But my point is that the mass core was able to increase because cooling ions were falling upon it taking with them electrons for neutrality, because some of the hot ions in the "intermediate" part of the star (NOT in the immediate neighbourhood of the WD inside the star but not the far part either) were cooling. If that mass had been already in the WD core all stars with very large masses would have become neutron stars or BH.

If too many fall in that's what happens. If enough of the total mass is expelled, one gets a WD. It has nothing to do with the cooling of the WD already in the center. But the way the different parts of the star cool is important.