Obviously ISO 10666 applies here...
Stephenson is to be saluted for not writing the obvious book in which either
a) the avout are some sort of theocracy, probably using science as magic to control / fool the non-avout and/or keep them at a lower level of development (see The Prince In Waiting trilogy by John Christopher, Heritage Of The Star by Sylvia Enghahl and probably numerous other books, juvenile and not)
b) another group is controlling things and keeping the avout and non-avout at a particular level of development.
Nonetheless, it would be nice if the whole thing made a bit more sense.
Clearly, the following may apply (beyond 'who cares')
1. It's not Earth, they're not human
2. Raz is very far from being an omniscient narrator
3. A closer reading may prove I am actually wrong here
Having said that the following leap to mind (10 to be going on with, not in order of importance)
1. Clearly quite a lot is known about what happened before The Reconstitution, e.g. somehow vast amounts of modern physics is known but somehow major parts of the history of the equivalent of our 20th century has been forgotten (the excuse is that it was lost on recorded media, but why doesn't that apply to the physics?).
2. What is the 'day gate' for? A day gate only makes sense if there is someone to interact with, which presumably there isn't because even the Oncers are supposed to be cut off for a period of a year.
3. There is no clear motivation for why the different timescales of the different maths came about. The motivation for putting scientists out of the way does not extend to putting them out of the way for differing time periods.
4. Indeed, if you think scientists were responsible for bringing about the Terrible Events (TM), wouldn't killing them all make a lot more sense? The 'we do not trust these people so lets put them out of the way where they can get on with stuff (especially for periods of up to a thousand years)' idea does not seem like the first one that would spring to mind. The same applies pretty much to leaving the 3 inviolates alone after the 3rd sack. There is of course an explanation i.e. they are minding the nuclear waste, but all things being equal wouldn't you decide that someone else could mind it under the circumstances. Hmm, there are people with scary powers, better leave them alone to mind the nuclear waste.
5. What is with the copying stuff out by hand? I can understand the not being allowed technology (except of course we have the bat-shit 'allowed some sort of computers as long as they don't have the root password' exception for plot purposes), but not allowing scientists moveable type seems a little harsh. Also, presumably, the reason for having the different maths is for some sort of societal stability over long timescales, and the ability to make decent copies would seem to be a big plus in that direction.
6. I am not sure I buy the willingness of the avout not to have children. (particularly since it is IIRC explicitly said that it is the male avout who are rendered sterile by the drugs, although there is no mention of post-Apert pregancies after a quicky in the Saecular world, and there is no suggestion that the drugs stop people wanting children anyway)
7. Nor do I really buy the rarity of Anathemas. I assume that it is implicit that there is a time period (e.g. between joining the concert and the aut of Eliger) where you can be booted out without a proper Anathema. But even so...
8. The ita. Probably because of the limited knowledge of the first person narrator, we don't really know how the ita work. They must get paid somehow given that they live in the saecular world, but how? And given they've got the technology why aren't they running rings round the avout given e.g. their ability to copy documents not by hand? Also, where do you get ita from during dark ages? Wonder if Stephenson did ever read Heritage Of The Star, the avout pretty much match the scholars and the ita the technicians. And they too aren't allowed children, but in the Engdahl they have to give them up for adoption.
9. Fundamentally the ability to survive regardless of civilisations rising and falling around them is the one I find hard to swallow. What about pollution, polluted water courses etc. (not sure I quite buy their energy self-sufficiency either but in both cases I suppose we could rely on some madeupeum not mentioned)
10. If Jules Verne Durant is French, why does Earth become Earth later on? Assuming we believe that enough Terrans are on the ship to support multiple languages.