> Can't all proofs be eventually broken down into their fundamental pieces and then it's clear as day if it's right or wrong?

You’d think so, but not really. There are mathematical structures which are unimaginably huge but have little if any reducible structure. For example, in algebra, one of the most basic structures is a Group. When trying to understand a group, one of the most important tools is to break a group into chunks using what’s called a “normal subgroup”. However it turns out that there are some absolutely enormous groups that are “simple” (ie have no normal subgroups). So, there is a set of 26 of these known as the “sporadic simple groups” that just don’t fit any kind of pattern. Proving results about these has proved very difficult because they can’t be broken down (they have no normal subgroups) and by definition just don’t fit any kind of other pattern. One of these, the “monster” group has approximately 8x10^53 members. So you have a set that is unimaginably massive and has very little internal structure as it is “simple” and so can’t be broken down further.

The proof that there are 26 of these sporadic simple groups is part of the theorem known as the classification of finite simple groups, sometimes known as the “Enormous Theorem”.[1] It took over 100 mathematicians nearly 50 years and resulted in hundreds of papers. Even with that many mathematicians involved, there were still errors and revisions needed to the original proof. Some of the original authors are gradually publishing a somewhat simplified version of the proof but it’s still a massive effort.

[1] https://en.wikipedia.org/wiki/Classification_of_finite_simpl...