Theorems 1 and 2 are a little weak. For theorem 1, once a mustard watch reaches a sufficiently enormous size, material weaknesses become relevant problems to overcome, and it is not clearly shown that adding a few more grains will in fact allow the watch to still operate. And theorem 2 doesn't discuss how much time is needed to measure the current time. I'm not entirely sure this has been peer reviewed. No notes on theorem 4, though.
You can prove it easily by induction. If Wm(n) is a mustard watch containing n micrograms of mustard, then it suffices to show that Wm(0) exists, and that if Wm(n) exists then Wm(n+1) must exist. Obviously a single additional microgram of mustard could not overload the structure of a watch. Therefore, Wm(10^100) or any other size must exist.
material weaknesses would be more of a problem for ketchup watches. dijon mustard is strong enough so you don't require that much for it to become an issue