Mathematics has got about to European Renaissance levels. Theurges and engineers are thoroughly familiar with algebra and trigonometry, but no one has invented Cartesian geometry or calculus.
Sociology, psychology, economics, and political thinking are all still just parts of philosophy; none of them has anything resembling the data-driven empiricism most people think of today when we talk about “social science.” Nor will they for centuries. Studying culture isn’t yet conceived as a science in the gameworld either.
Economically, the Karagond Syntechnia guildmasters are the only people allowed to act in a bank-like way – making formal large-scale loans to the army, the priesthood, and the great magnates (mostly noble, mostly Karagond) of the Hegemony. For everyone else, small-scale, high-interest credit is widely available from local merchants, or the local lord/lady.
The urban areas along river/canal corridors have a pretty monetized economy for everyday transactions, relying on silver and gold struck in the Hegemony’s fifty-odd mints. In more rural areas, people tend to get by without much coin, using other locally variable means of keeping track of mutual obligations.
For transactions in between the “day-to-day” and “fund an army” scale, the Syntechnia merchants use their formal knot and seal system to avoid the need to constantly move hard currency around, and also to keep certain guarantees and transactions out of view of the Telones, who aren’t taught the whole knot-language. For the most part, that’s the Hegemony’s equivalent of promissory notes and bills of exchange.
The major language groups roughly correspond to the provinces, though all of them have a lot of distinctive dialects. For Wiendrj in particular those dialects vary enough that they should really be considered as about two dozen different languages, with local dialect variation almost valley-to-valley through the mountains. You’ll get more details on the Shayarin language in Ch 2 when I’m done with it. The Koine (common) Karagond language unites the provinces.
There isn’t really “an” imperial bureaucracy – none of the gameworld empires have developed China’s repertoire of control. China was relatively rare in our world in its reliance from a very early stage on a centralized, professionalized civil service, where other empires tended to coopt local elites to get things done in ways suited to the local culture and context. The Hegemony sends out its priests, Alastors, and Telones – three very different institutions in scale, literacy levels, and capacity – but also delegates a lot of practical authority to loyal aristarchs. The balance of functions between the local elites and Hegemonic institutions varies wildly from place to place, as done the extent of record-keeping.
Theurgy helps with the construction and maintenance of deep wells, aqueducts, and big sewer drains, but the big cities are still for the most part miasmic and unsanitary. Only the intervention of healing mages keep the urban population from collapsing due to epidemics.
This is going to be one of the things that correlates with how big you want your Game 5 realm to be. Satisfied with a small state? You can do more with whatever state capacity you’ve managed to preserve/create, including making your population more legible to you and exerting more control over institutions. Want to try to build a new big empire on the ruins of the collapsed empire? …I don’t want to prejudge the emergent results of Game 5 now, but even if you’re pursuing a low-anarchy strategy, I think it would be pretty nearly impossible to come out of an imperial collapse (when plenty of others will be stirring up chaos!) with a net gain in state capacity and the legibility of your population on a continent-wide scale.
“A war of ideas through the illicit printing press,” I mentioned in the thread’s first post. It bears emphasising that “press” in the gameworld isn’t a metonym for “newspaper.” It’s just the literal machine that lets you do mass printing. Given low literacy and tight control over printing techne by the priesthood, there’s never been a tradition of newspapers in the gameworld.
If you get hold of a press in Grand Shayard, you could fire off a lot of one-page broadsides and chapbooks to reach the yeomanry or free urban poor, maybe short pamphlets of 5-10 pages if you’re trying to sway more literate demographics like the aristos and priests. But no one’s going to be founding the Grand Shayard Times, I’m afraid.
Jonglers, musicians, dramatists performing plays, rhetors performing speeches, acrobats and contortionists. And probably more I haven’t thought of yet. 
Your realm’s post-collapse murder rate (and crime rate more generally) will vary based on total anarchy. A low-anarchy strategy will be helpful for lots of state-building purposes, but inconvenient for any MC who wants lots of criminals to send to the Harrower. A high anarchy approach (that avoids going right off the anarchy cliff) would probably leave you with a rate like the most dangerous non-failed states in our world, say 40 or 50 murders per hundred thousand people.
Let’s reckon that you find someone halfway credible to blame for 60% of those murders (if you wanted to actually solve them beyond a reasonable doubt in a high-anarchy early modern state, you’ll end up with a lot fewer executions). That would yield about 67,000 Harrowed murderers per year across the whole Hegemony at current population levels. If you want to prioritize keeping the palace in the air, that’s one-sixth to one-fifth of your murderers’ blood harvest gone every year before you’ve ripened a single crop or warped a single Plektos.
Obviously a state of the kind you have in mind won’t have murder as its only capital crime. But I think the back-of-envelope estimates above suggest that it will be pretty hard to run a Harrower-state off executions alone without slipping into one or another dystopian outcome. (“100% of murders in my jurisdiction result in someone getting convicted and Harrowed! And 100% of the attempted murders too! That guy was just thinking about killing somebody!” “We recently expanded the death penalty to include ‘loitering with intent’.” “Out here, it’s a capital crime to be Karagond.”)
So, yeah, I think any MC really committed to keeping the palace in the air is going to need to adopt a Harrowing strategy that goes well beyond anything deserving the name of a justice system.
Red with black accents, especially a black accented omphalos symbol, would fit just fine for them. I’m happy to make their helmets black. 
Only K and S will show up regardless of your G1 approach, not Edwer/Etthena, or Bethune/Bjel and Korszata. I mean, I reserve the right to change my mind and make things more complicated for myself any time… but I don’t think I’ll have them join you for anything other than a high G1 score.
Certainly not for the end of G1, by which time I think the most committed high-anarchy runs have been racking up scores in the 60-80 range. But it’s high enough, given what it means for how you’ve played to that point, to turn off Simon/Suzane in Ch 2.