Vatican Rejects Intelligent Design?
I couldn't agree more fully with your opening paragraphs stating what science is. Congratulations, you are one of the very few people who actually know what science is and what it is not. However, you don't seem to know what irreducible complexity means.
Irreducible complexity is applied to natural selection, not any physical process. For instance, without gears a watch is completely useless. Without hands, a watch is completely useless. Without the face, a watch is pretty much useless. Now, think about a watch as an eye or some other ridiculously complex organ (or molecular machine, such as ATP pumps - slightly more difficult to imagine). The individual parts of the whole are useless, in and of themselves. However, it is only when put together in a very specific and total way that any of the parts have value. There's no reason to have one without the other.
Irreducible complexity is simply the claim that evolution does not allow for the development of organs or molecular machines that are all or nothing. In evolution / natural selection, everything must be reducible in nature. It is intuitive for arms and fins and wings to behave under evolution / natural selection, but less so for parts of organs / molecular machines that are co-dependent and have no individual function.
It's a difficult topic to explain, hopefully that made a bit of sense.