Nothing you said was incorrect but I'm afraid people might take away a misconception. So, the adders you find in a modern processor are probably going to be a

Nothing you said was incorrect but I'm afraid people might take away a misconception. So, the adders you find in a modern processor are probably going to be a carry-bypass adder or something like that. For a constant factor more transistors you can complete your addition in sqrt(n_bits) time rather than n_bits. There are also tree adders which can can perform their calculations in log(n) time but take n^2 transistors but nobody uses those. Most modern processors have pipeline stages where you have one transistor driving 4 other 10 to 30 times or the equivalent. You can do that even with 64 bits in a clock cycle with a carry-bypass but not a serial adder. https://en.wikipedia.org/wiki/Carry-skip_adder https://en.wikipedia.org/wiki/Carry-skip_adder