[talanum1 (OP)]

Here is the proof:

Line number       Statement                                               Reason
1.                        Assume a Black Hole has a matter density singularity.   Assumption.
2.                        We know that a Black Hole has a temperature.              Empirical knowledge.
3.                        Having a temperature implies the Black Hole has
                           particles with an average Kinetic Energy.                  2.
4.                       This contradicts 1, since there is no space for them to
                          move in.                                              1, 3.

[Kryptid]

We know that a Black Hole has a temperature.

The temperature of a black hole is due to the Hawking radiation at the event horizon, not the singularity. Your "proof" is no proof at all.

[Bored chemist]

Here is the proof:

Line number       Statement                                               Reason
1.                        Assume a Black Hole has a matter density singularity.   Assumption.
2.                        We know that a Black Hole has a temperature.              Empirical knowledge.
3.                        Having a temperature implies the Black Hole has
                           particles with an average Kinetic Energy.                  2.
4.                       This contradicts 1, since there is no space for them to
                          move in.                                              1, 3.

2 is dubious; 3 is wrong.
The temperature of the radiation from a black hole is the temperature of that radiation, not the temperature of the BH itself.

Since - so far as it has a meaning- the temperature concerns particles which are outside the EH of the BH (and potentially indefinitely far from the BH itself, you can't justify those statements.

If you had a BH the mass of the Earth then the radiation of which you were measuring the temperature would be coming from the EH. That's about 88 mm from the BH itself.
Now, I accept that 88mm isn't very far, but the point remains it's not zero.