Traius, the key word in what I said is 'higher'. You and Seraiel (et al) chop and rush for the resources, and I have no problem with that. You get more cap than you lose from chopping the forests so you are ahead of having just kept the forests. I have no doubt that you do get more cap than you lose so you come out ahead. But if you keep the forests and claim the resources then you have a higher cap than you get from just claiming the resources. As long as X and Y are both positive X+Y will always be greater than either one.
As I said, this higher cap shows up too late to matter to Seraiel, which is fine with me. It shows up too late to matter to you, which is also fine with me. I just enjoy having the outcome of the game still be in doubt in the industrial era so I play differently than you. Thanks for breaking it down, but I actually understood the first time.
Even that is wrong. If you simply chopped those forrests and took cities with the units from them, you would have had so much more research, that you could have easily gotten Future Tech already, and Future Tech also gives

and

.
What you're also totally not mentioning is, that

and

caps lie above 20 only from resources and good or even key buildings. Getting cities beyond a population of 20 is totally ineffective, because normal cities do not even have more than 15 tiles they could work, maybe 1/3 of them even being good.
Thanks for your analysis Traius. I am new to looking at civ from such an economics point of view but I think you have missed some things:
1 chopped forest is approx equal to 1 axeman. But it usually takes the loss of more than one axeman to take a city in an axe rush. I would say 2-3 is about right. Granted a city may be worth more than 1 health resource but it might also be worth none. With the right infra a health resource might be worth 2 health though and that does go to all cities as you point out. But its wrong to imply 1 chopped forest = 1 captured city.
- Taking the city incurs a maintenance cost that leaving a forest does not have.
+ Taking the city incurs a xp bonus to the units which survive.
- Leaving a forest allows it to be worked for 1-2 base hammers. This is very marginal as working a forest is not usually a cities' best option and should probably be whipped but may be useful in production poor cities which are often FP heavy cities which are health critical.
? Leaving the forest does not necessarily mean not taking the city, as Timsup2nothing pointed out, chopping will get it sooner and therefore probably cheaper. So the comparison is often that finite but early cost saving from chopping (plus benefits received from city for the extra turns) vs the ongoing but delayed benefit of not chopping.
? A forest which is not chopped can always be chopped in the future and may, in fact, produce more forests by being left. Granted the value of the 30 hammers decays rapidly with time. I can imagine a situation where chopping that forest boosts the production of a longbow by 1 turn and saves the city. (ok I'm stretching)
Any comment and criticism of my analysis is greatly appreciated, this is something I want to get better at. Cheers.
You're right, 1 Forrest does not always transfer to 1 city, it can be less, it can be even more. If you i. e. play with Rome and build Praets against a Target having Archers (not even an unusual case) , 1 Praet can transmit into maybe 3 or 4 cities, as the chances of a Praet against an Archer are somewhere near 80%. Anyhow, it doesn't even matter if 1 or 2 Praets are lost, if the city is lets say size 5-6, because then it can replace the reinforcements itself, and as the city probably will have forrests again, the advantage of having taken it even further multiplies. This effect is known as "snowballing" in the strategy-players-community, because the snowball launched grows into a lawine and the earlier it's launched, the greater the lawine. And don't forget, that a conquered city also gives Gold, fueling the research, which already will be higher than normal because of the additional traderoutes and worked tiles.
P.S.: Lumbermills are, at least for me, no argument, because a Workshop with SP or even Caste + SP is better than one, as 1

= 2

.