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Most of the time I have my Marshal train troops, my Steward collect taxes, and my Court Chaplin research Cultural Technology. Except I don't actually know what that Technology Spread effect is doing precisely.

When a council member improves the technology spread rate, does that improve how fast the county accumulates tech to the next tech levels, or does that make the county spread its increased levels to nearby counties that have lower tech levels, or what?

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The CK2 Wiki has an excellent and up-to-date article explaining technology. It has this to say about technology spread:

Technology spreads between provinces that border each other, as well as between provinces in the demense of the same ruler even if they do not border each other. The rate of growth depends on the degree of the technology lead in the more advanced province. The Spymaster's Study Technology mission also lets technology spread from the target province. The other advisors' research missions also increase the speed of spreading, as do the School and University buildings.

So by sending your Court Chaplain (or Steward, or Marshal) to research tech in your capital, you increase the rate at which the capital increases in cultural (or economic, or military) technology and therefore increase its lead over provinces neighbouring your capital, which in turn increases the rate of spread of tech from your capital to its neighbouring provinces and non-neighbouring provinces in your personal demesne.

This is a good thing! Your capital is where tech will increase the fastest, so you want to encourage those advances to spread to other provinces in your realm. If your capital borders directly on provinces outside your realm (for example, Krakow in Poland borders Rus provinces and Hungarian provinces), either accept that you're helping your rivals somewhat, or move your capital to a more central province.

If you want a particular province to catch up faster instead, sending your councillors there will multiply the effect of spread from its neighbours.

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Oh, google kept giving me a different wiki, and the article they had used a picture for the tech categories that wasn't even the current set of categories, so I wasn't trusting that much. This seems better. –  Lokathor Aug 2 '13 at 8:49
    
@Lokathor Yeah, there being two different wikis is a bit of a pain. (I think Google likes the other one more because it's a Wikia wiki.) I find I get the most out of cross-checking them both. –  SevenSidedDie Aug 2 '13 at 14:02
    
...Court Chaplain to research cultural tech in your capital, you increase the rate at which the provinces neighboring your capital "catch up"... - I don't believe that statement is correct. This would increase the rate of spread from neighboring provinces (+demesne & spymaster) to the capital , not the spread from the capital to the neighbors. –  Affine Aug 18 '13 at 3:23
    
@Affine I don't believe it works that way. Chaplains increase "spread", which is a downward/outward thing. Increasing the spread of a province increases how quickly technology spreads from it. Putting the Chaplain in my capital seems to makes a difference, which wouldn't make any sense when my capital is way ahead of all neighbouring provinces. –  SevenSidedDie Aug 18 '13 at 14:03
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@SevenSidedDie See here. First is no council in capital. Second is counciller added. Third is capital moved to an adjacent province (since as far as I can tell, you can't get tooltips for provinces that are not your capital). As you can see, the councilor bonus increases spread to the province he is in, not out of the province he is in. Also, see the wiki formulas slightly down from your quote. –  Affine Aug 18 '13 at 14:55

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