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Always chop forests on mines, always chop before placing a district on a forested tile. For forests on flatland, it depends. Definitely chop for the key wonders like Pyramids, Oracle and Kilwa and chop for important infrastructure but if the city is really production starved, leaving some forests for lumber mills can be fine. Lumber mills just unlock a bit too late for my liking.

Chopping for settlers is very useful. It's perfectly fine to gimp one city a little to churn out more cities faster. Magnus chopping settlers is such a huge production boost for your game, the earlier you place new cities, the earlier you place down their districts and the cheaper those districts get. But I will usually wait to start the chopping till after Colonization is plugged in and the Ancestral Hall is built to maximize the efficiency of the chops. I usually try to build 2-3 settlers before this, and might chop some of them, but prefer not to.

Also, chopping out Horses for a Horseman rush is one of the strongest moves you can make if you have an attackable neighbor. Again, gimping some cities to gain more land usually ends up worth it in the long run. In civ 6, more cities generally beats out better but fewer cities. Legion chopping is also a classic strategy as Rome, legions chop out more legions and snowball to kill your neighbor.

I think the main flaw in your idea is that early production snowballs really hard in this game. Lumber mills come late, having a really strong city at that point in the game is not quite as useful as the early production boost of chopping. Later in the game, you can easily create 3-4 powerhouse cities by centralizing trade routes. Internal trade routes pre-tier 3 government will provide decent growth and production and if you have a commercial/harbor in each city (which you usually should), you can have 14-20 trade routes easily. With democracy or communism, each trade route can be 8+ production, so most of my late game production comes from trade routes, if I'm not playing a civ with good industrial zone bonuses or a strong coastal civ with lots of coastal cities.

But for the most part, I do avoid chopping flatland forest unless absolutely necessary. Forests on hills however, should absolutely be chopped every time. Mines are just better, and you can get both the instant production boost and a strong productive tile. Saving some chops can also have some value by the way, you don't need to chop immediately, and chops scale in value with tech and/or civic progress. Chopping in the final steps of a science victory is very common (buy spaceport, chop in projects).

One other thing you haven't mentioned is food chops. Chopping food resources, while unrelated to this forest issue, is critical in getting newer cities up to speed, and to get cities to pop 10 for rationalism.
 
Rain forest lumber mills come pretty late. Lumber mills on woods come in Classical (oops, early medieval). One key reason not to chop everything in sight is to avoid droughts.
 
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I'm confused what you mean by bonus food. Lumber mills provide no bonus food to the tile, nor do woods.

Lumber mills are:
+1 for the woods, +2 production for the base mill, +1 on steel (way too late and irrelevant), +1 on cybernetics.

Mines are:
+1 base, +1 on Apprenticeship (which is usually before I even unlock Lumber Mills), +1 on Industrialization (well before Steel and actually somewhat relevant), +1 on Smart Materials (completely irrelevant)

So Mines are basically always better than Lumber Mills when not taking the woods bonus into account and the woods bonus alone will almost never equal the value of the chop, even subtracting 2 build charges worth, as long as you use Magnus.

Lumber Mills also unlock at Machinery, which, unless you're doing a crossbow rush, is quite late to be waiting to improve your Hill tiles.

Rainforests provide 1 food, but you're almost always better off chopping Rain Forest because the food chop really accelerates city production. And waiting for Mercantalism to improve them is silly, the game is more than half done by the time you hit Mercantalism.

You have to weigh the yields against when you actually unlock them. Everything about Lumber Mills is slightly too late, and they are only really good on flatland forests or for rainforests if you're Brazil or went Sacred Path + Work Ethic.
 
it's always going to depend somewhat on local geography and what you're trying to do. yes rainforests only give +1 food, but if you have a lot of them, that can definitely add up. Chopping might not necessarily be the best idea...if the city is otherwise poor in production or food (in order to actually work the production) then after chopping is done the city will be kinda useless.

granted early production is likely to snowball and be a good idea most of the time. But not ALL the time IMO.

also no one ever seems to mention the environmental protection leaving occasional forests provide. I kinda enjoy the minigame of preventing droughts by leaving an occasional forest. Of course if you turn off disasters that is not relevant but for some people it might be.

I would venture to guess lumber mills are used more often by more casual roleplayers (they are pretty and especially attractive to roleplaying as "peaceful good guys") and less often by hardcore strategists trying to get the fastest win times.
 
If you lean towards chopping early, you can always replant the woods and then build lumber mills after getting Conservation. That admittedly is pretty late in the game, but if you really want to convert a flat land city into more of production power-house that is an option.
 
For me, the question of chopping isn’t centered around “if” but “when”.

Theoretically, to maximize your civ, you’d want most every “hard” resource(woods, stone, crabs, etc.) chopped by the time you win.

Typically, earlier chopping is better because, as in the real world, the value of production/money is discounted over time.

However, some things in Civ are worth saving chops for because they should be completed quickly/immediately after they become available. For example, you may save up crab/maize chops so you can buy a spaceport. It’s also good to save chops to immediately finish the moon landing for the culture boost. You could make the same argument for any key wonder such as Cristo Redentor for a religious tourism victory.
 
Harvesting is the closest we can get to transferrable instant production from one city to another.

Thus, the answer is, yes, harvest whenever you want now, rather than X turns later.
Every resource loses value from turn 1 onward. Exactly how much, that's really hard to calculate, but it does.
 
Harvesting is the closest we can get to transferrable instant production from one city to another.

Thus, the answer is, yes, harvest whenever you want now, rather than X turns later.
Every resource loses value from turn 1 onward. Exactly how much, that's really hard to calculate, but it does.

Chops do also increase in value as you research techs and civics, so waiting a little is fine, especially if you can increase the number of builder charges at the same time, or time out a critical wonder like the Pyramids.
 
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