One of the paradoxes of evolution is the evolution of proofreader enzymes, in light of the theory of mutations and selected evolution? The proof readers enzymes correct any mutations that result during DNA duplication. Natural selection, by evolving proofreader enzymes was trying to get rid of the randomness within DNA replication, or else why evolve proofreader enzymes designed to fix these errors? The proof reader can see the improper base pairs by their energy signal; higher potential than proper base pairing.
The work around is the proofreader enzymes are only active when the DNA is duplicated, so they will still miss environmental changes on the DNA between DNA duplication cycles, such as by viral insertions or other environmental changes. This suggests a more real time connection to the environment fueling the selected change.
In terms of irreducible complexity, modern humans cells will not function without the organelles called the mitochondria. However, the mitochondria are able to function by themselves.
Right now the theory for mitochondrial in modern cells is two existing things merge, but now only the mitochondria are self sufficient if separated. The question is how did the majority of the modern cell form, before the mitochondria entered, and why is it now no longer self sufficient? And what is the proof? Or did the Mitochondria act as a hub, allowing the rest of the modern cell to form as an extension of itself? The irreducible complexity arguments infers the last theory. It can be applied to theoretical biology.
In terms of humans, only the female passes the mitochondria forward; eggs. The male provides the other half of the DNA. Sperm DNA get the mitochondria support, through it own insertion into the nucleus. The mitochondria energy output is key to the cell cycle. This suggests the main DNA was not very functional until it had the extra energy supply via the Mitochondria. Virus do a similar but smaller dance than sperm cells. Virus are irreducibly complex but can get beyond that by piggy backing on a viable host, adding its own DNA to the host.