The flaws, on the whole, tend to be essentially that although world-building is an important feature of the book, the world-building is not actually that good in the sense that the answer to the question 'why do they do x?' is invariably 'transparently due to the exigencies of the plot'.
1. Whilst clearly Erasmus as first person narrator has to be in the middle of the action, it is a large coincidence that 4 of the big players are from St. Edhar's - Orola, Ala, Paphlagon and Jad - and that two of these (Orolo and Ala) are important people in Erasmus's life. We might just about say that Orolo and Paphlagon's significance is connected but it is a bit of a stretch, perhaps.
2. Related to 1, there is no reason given at all as to why Jad is evoked out of all the thousanders with wibbly-wobbly-quantumy-wantumy powers in all the maths.
3. Also related to 1, we are told that Ala is evoked because she is really good at organising things. So, to summarise, the world is under thread from aliens with nukes, so the mathic powers choose a 19 year old who is known to be really good at organising things. Based on, er, what, precisely? Now, clearly we have to make allowances for the mathic world operating on very different lines from the world as we know it, but I still come back to 'based on, er, what, precisely?'. Beyond this, whilst St. Edhar's is essentially a monastery without God and with sums, Tredegarh where the convox is being held seems on the other hand to be run much more on the lines of an Oxbridge stylee University e.g. fairly hierarchical and pompous etc. and how likely is it that one of them would choose a 19 year old known to be really good at organising things?
4. It can, of course, be explained that Raz doesn't really understand what the Ita do because there is a big taboo against him knowing anything about them, but it is hard to see that the ITA make any sense - as a parody of grumpy sysadmin types it is a moderately good joke, but they really don't have very much to do. The concent is essentially low-tech apart from the bolts and spheres, which don't seem to need any maintenance, so all the ITA seem to need to do is
a) maintain the 'needed for reason of plot' technology used to take long exposure photographs of the sky
b) replace bits of the clock whenever this is needed (not very often)
c) send emails for those avout allowed to send and receive emails (the wardens, presumably)
Hardly seems worth a separate caste. Also, how do they manage to get the computers etc. when society breaks down?
More importantly, since they clearly DO study computer science including more metaphysical stuff like 'The Aboutness Problem', in what sense is there any convincing break on them using their leet Comp-Sci skills to bring about the singularity, or whatever?
5. Although the equivalent of our times on Arbre are called the Praxic age, there seems to be a decent amount of Praxis in the time the novel is set - they launch satellites after all, and have GPSs. So is the idea that 'thinking about hard sums' would be such an alien concept to people plausible? Clearly it is supposed to be but it is not conveyed very convincingly, lots of telling, not much showing.
6. Other than some sort of numerical symmetry it is not quite clear what the point of the 100 year Maths is. The only functional difference between an avout in a hundred year math and one in a a thousand year math is that the 100 year avout gets a 50% chance of being an adult when one is allowed to wander around for 10 days in a world one is completely unfamiliar with and does not speak the language of, the 1000er forgos this dubious benefit but gains the ability to use wibbley-wobbley-quantumey-wantumey magic to live longer, manipulate the fabric of the space time continuum
7. The wardens are allowed to oversee the 100 and 1000 year Maths so presumably if they are at all observant some of the wardens are in on the longevity, and wibbley-wobbley-quantumey-wantumey secret. Obviously they have to keep quiet about this. Maybe they do, but this seems implausible. Discuss.
8. Whatever the purpose of the 10, 100 and 1000 year Maths is, surely it is a big violation of that to allow people to 'go through the labyrinth' to the next Math along to do harder sums. Even if we postulate that the progress of 'sums' is very slow because of 3000 years of extra development, if the point of the different duration maths is to either (a) slow down progress [which doesn't really work, since there is nothing in the book that says people can be ordered into the next math along because they are making too much progress] and/or (b) avoid the vagaries of fashion in whatever they are studying, then the ability of someone to pop up between the 100 and 1000 year aperts, this is going to blow a big hole in the reason because the person moving up is not going to be able to conceal their knowledge of new development. Imagine, for example, that the apert of a 100 year math was in 1890, but someone from say 1950 was able to go into the 100er math. How would they be able (even if they wanted to) to not bring over any knowledge of mathematical / scientific progress since 1890? [As noted at the beginning of this section, this is an exaggeration as we presume progress after 3000 years of mathematics is going to be slower, but if it isn't an issue, why bother with the different maths?
9. As I have said before, the 'day gate' doesn't make sense - what is the point of a day gate if avout can't interact with those who come through it? I think we have to assume that 1-ers can really interact with people.
10. If it becomes apparent that avout can manipulate space-time thanks to wibbley-wobbley-quantumy-wantumy stuff, what is the upside to the Saecular world of allowing them to be left alone to get on with it after 70 years?
Discuss. Or not.
Also, I thought REAMDE (a.k.a. GOATSE) was crap.