The conclusion does not seem to follow from the premise. For one thing, it ignores both the possibility of mathematics 'adopting' the scientific method after it was formalized, and the possibility of mathematics using the scientific method without naming it as such.

However, even granting your conclusion that mathematics does not use the scientific method, it still does not follow that 'formal science' is merely science without the scientific method. In other words, if something does not use the scientific method but does fit all the other criteria of an empirical science, that does not automatically make it a formal science.

Generally, formal sciences use a method similar to the scientific method except that theorems 'must be proved', instead of 'should not be disproved with a counterexample even after a certain amount of testing'. The 'must be proved' requirement is actually equivalent to 'should not be disprovable with a counterexample regardless of the selection of testcases', so is a stronger requirement.

Following that line of reasoning, one could argue that formal sciences are the only true sciences, and that empirical sciences have been accepted as also-sciences merely because nothing better than the scientific method is available for the respective subject matter.

I don't subscribe to that point of view, but I do not think that 'mathematics is not a science' can be stated as a matter of fact.

Well, mathematical research usually initially involves observing a pattern in abstract models, crafting a hypothesis, and testing the hypothesis in specific abstract models to find a counter-example. So to that extend mathematics uses the scientific method.

Of course, afterwards you need to find an actual step-by-step proof of your conjecture from commonly accepted axioms: it doesn't suffice to merely fail to find counter-examples. By necessity, empirical sciences are less strict that way.

Math is considered to be a formal science (as opposed to an empirical science). The general term 'science' applies to both subclasses, so math is a science.

Actually, the judge is correct. Some people seed more, some people seed less, but on average the number of uploads for each bittorent participant is equal to 1.

The reason is, for any given file distributed through bittorrent, the average number of uploads or downloads per person is each equal to the total number of uploads or downloads, divided by the number of persons participating. Since each kilobyte downloaded is uploaded by someone else, the total number of uploads and downloads are equal. So the average number of uploads per person has to be equal to the average number of downloads per person. And for any participant, that average number of downloads is 1.

I'm ignoring the possibility of incomplete downloads, blocks that needed to be re-downloaded, or the fact that the original seeder didn't need to download the file, but those are fairly minor factors that will not substantially alter the result.