@peregrine23: Thanks a lot!
Somehow I missed the point about different levels of base production as a level of population. Makes it a bit difficult to reproduce numbers without knowing the formula...which I think is an important part of testing, but whatever.
So my original idea that approval bonuses apply to base production (because the tool tip says "production") is valid, while those labelled manufacturing apply after base production is determined, to wit,
Base production * fraction allocated * (1 + sum of various manufacturing bonuses expressed as fractions).
I tried to make everything work by supposing that perhaps the tool tips were mislabeled as to production/manufacturing. Naturally, that didn't work either!
And of course I got very close to the right answer for colonies whose pop is very close to 10, because the 1:1 prod:pop is very close, which confused the issue more.
To continue my next goal is to confirm the calcuation project production. But until I find the formula for the above, it's pointless.
I will post a link if searches are productive!
Thanks again.
EDIT: I wanted to provide a link that really generated most of my confusion, as it clearly stated 1 pop = 1 production. But it was early and has been changed.
https://forums.galciv3.com/452497/page/1/#3449677
Still looking for the update.