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Which means that the first two assumptions are wrong, leading us to conclude that there exist minimal perfect programs for every specification.

No. If we negative dorfsmay's premises, it still does not lead to your conclusion.

There's a bug in this zero-byte program: the other program might not leave its memory in a correctly runnable state when it shuts down.

Q.E.D.!!!

The issue is whether "any program out there" is taken to mean "any existing" or "any possible". It may be true that no programs actually exist that couldn't be shorter or more correct (though I doubt it), but one would reach a minimum length and bugginess eventually.

IEFBR14 was a one-byte program from IBM that had a bug. The history of the program makes for some cool reading: http://en.wikipedia.org/wiki/IEFBR14

Please note, this is a joke, don't argue with it too seriously. Also it's not my joke, it was floating around in usenet days.