Your grasp of this simple fact is clouded with your perception that tooth size or count means anything in ratio.
It doesn't matter whether the outer crank sprocket is 6" in diameter or 12", it will have the same ratio if teeth count are the same, providing the same output.
The teeth and pitch dictate how fast the sprocket turns. You need to have a better look at your own "grasp of this simple fact".
Diameter is directly related to teeth count and pitch, and tooth count+pitch are what influence ratio in the case of sprockets.
You can decide to just use diameter as a figure, but you
cannot discount teeth and pitch when discussing sprockets.
Divide the drive pulley diameter by the load pulley diameter.
The result will be the pulley and belt ratio.
For example: a 2-inch drive pulley and a 4-inch load pulley yields a pulley and belt ratio of 1/2. This means that for every 1 turn of the drive pulley, the load pulley will make 1/2 of a full turn.
Pulleys are not sprockets. They do not have teeth. Diameter means everything to a pulley. Pitch has no influence on them.
It simply does not mater what connects the two sprockets unless you think somehow a 2" sprocket to a 4" sprocket can somehow make something other than 1:2 gear ratio reduction by the chain or belt that connects them, they are one with each other regardless.
I never said otherwise. My point is that keeping the jackshaft output tooth count the same (here let's say 10T), one can change the pitch
of the chain to dictate how large of a reduction sprocket (here let's say 56T) is desired. The output sprocket will be a different diameter
as a result of pitch (1.59" #41, 1.19" #35, 0.79" #25) and so will the reduction sprocket (8.91" #41, 6.68" #35, 4.46" #25).
You claimed you could do the same thing with 410 chain that I did with #35. You simply can't, because ½" pitch has limitations. 8T is available, but hard on chains.
All you can do is engineer a second reduction system or run a huge outer crank sprocket.
Your point is with smaller chain and more teeth you can get a sprocket diameter in-between what a tooth on 410 has, but it doesn't change the actual reality of gear ratio, that is the inside U diameter of the sprocket the chain rides in.
Yes, 410 sprockets have longer and fatter teeth, enough to add what, ~2mm to the sprockets outside tooth end diameter measurement, or are you saying the smaller tooth size is enough for a 1/4" reduction in overall outside diameter sprocket size which sounds pretty close?
I see that, I just want to make it clear that the inner U circle diameter of sprockets determines ratio, not what connects them.
For a guy who claims "Your grasp of this simple fact is clouded with your perception that tooth size or count means anything in ratio."
you sure do a great job painting yourself in ultracrepidarian colors.
I was one of the first people to shift-kit a 4G back in 2010, and I've put nearly 10K miles on it since. I've played with a _lot_ of gear ratios.
It is of course your right to think I don't know what I'm talking about. But I've hashed it, rehashed it, done it, and I'm wearing the damn t-shirt.
Good day to you.