I think it's likely the reduction is based on the base time required at the point the advanced scout is triggered and not necessarily the figures you're seeing in game which are the modified time.
I think you're mixing together two different concepts with this sentence. The second part about modified time is exactly the thing I'm investigating. By comparing two cities with the same number of scouts at the same ring level, but with 1 advanced scouts research different, I'm trying to determine how the base time is modified into what is displayed in game.
The first part though, about the timing of when you activate the Advanced Scouts research being a factor is possible. It seems weird to have programmed it this way, but I could imagine someone deciding to just keep track of the previous scout cost and add an amount driven by a formula based on province number and ring level. Here's a simple example of how they might look (holding ring level constant) assuming a 25% reduction on Advanced Scouts and adding 10*Ring*Ring every province:
Province | Ring | Advanced Scouts @ 20 | Advanced Scouts @26 | Advanced Scouts @ 33 |
19 | 2 | 760 | 760 | 760 |
20 | 2 | 610 | 800 | 800 |
21 | 2 | 650 | 840 | 840 |
22 | 2 | 690 | 880 | 880 |
23 | 2 | 730 | 920 | 920 |
24 | 2 | 770 | 960 | 960 |
25 | 2 | 810 | 1000 | 1000 |
26 | 2 | 850 | 790 | 1040 |
27 | 2 | 890 | 830 | 1080 |
28 | 2 | 930 | 870 | 1120 |
29 | 2 | 970 | 910 | 1160 |
30 | 2 | 1010 | 950 | 1200 |
31 | 2 | 1050 | 990 | 1240 |
32 | 2 | 1090 | 1030 | 1280 |
33 | 2 | 1130 | 1070 | 1000 |
34 | 2 | 1170 | 1110 | 1040 |
This would be very easy to test with the glut of newly minted Chapter V / VI cities from the low-level account migration. Take two cities ready to advance to Dwarves, have one scout 10+ more provinces than the other before entering the new chapter, catch them up to be level in terms of provinces, compare the timer values. For what it's worth, I don't think my data above confirms or rejects this idea, by my memory they advanced around similar province counts, and I don't have the actual number.
Assuming you are overscouted and I am not and that we both research the same Advanced Scout Tech.
If my scout times are 20 hours, I would get a 5 hour reduction. If your scout times are 26 hours you get the same 5 hour reduction. So I got 25% and you did not but we both got the same reduction
If I'm understanding this right, you're suggesting that you think the Advanced Scouts reduction is a flat value and is not proportional to things like provinces scouted or anything. Hey I could be wrong, but I can find no evidence of that in the data. If you can find some, please let me know!
It should be returning the same number if the arguments make in the OP are correct. The fact the samples show numbers that are not the same suggest it's not correct. Which means something's off. Not by much. The calculations in the OP I believe are distorted by all the samples having scouted more provinces than required for the completed provinces necessary to advance to the next chapter.
As above, it's possible that you're right about the timing of activating Advanced Scouts. But, the small discrepancies in my reduction numbers above (eg 20.59% vs 20.49%) are almost certainly cause by rounding issues. The underlying clock for scout times (and everything in the game?) is in seconds, but the displayed value is in hours:minutes. If I had access to the seconds value, I believe those differences would all but disappear.