Which of "couldn't", "can't", and "doesn't" for 13/13/13?

I recently came across an English learner saying that 12/12/12 is the last time such a repetition of day/month/year will occur in the Gregorian calendar for a long while, "because 13/13/13 wouldn't exist."

I don't know whether "couldn't", "can't", or "doesn't" would the best word to use, from a prescriptivist perspective.

I seem to recall that "can't" would be suitable for a real object that is unable to do something:

My car can't start

Or you could use "don't" for something that's possible, but doesn't happen:

I don't travel via the tollway.

But the problem is that 13/13/13 is not like a car. As well as being an abstract noun, it's also a logical contradiction. Does that mean I should use a subjunctive? Assuming that "couldn't" and "wouldn't" are subjunctive, this answer seems to suggest "couldn't" would be a better fit than "wouldn't", as it's an "Equivalent of "can" (possibility)".

Also, what would be the most common expression from a descriptivist perspective? Based on a google search of "13/13/13 * exist", it seems to be "doesn't" - does that sound plausible?

Just going based on what sounds right to my ear:

doesn't -- Preferred if I was just trying to state a simple fact. "13/13/13 doesn't exist". Much as I might say: "Little green men on Mars don't exist."

can't -- Preferred if I was trying to point out how the calendar system prevents such a date. "13/13/13 can't exist because it's an invalid date!"

won't -- Works as long as the date is in the future but sounds a little clunky. Wouldn't make sense to say this about 13/13/13 once 2014 rolls around, even though the date is equally invalid then as it is now.

wouldn't/couldn't -- Sound too wishy-washy, like they're entertaining the possibility that maybe the date could or would exist under the right circumstances.

Given your specific example, I think any of the first three could work, but I'd probably use doesn't.