Hmm, my first attempt at making a primarily mechanical adder[1] failed thanks to reverse power flow. So that was what the strange arrangements with double lines in other peoples' designs were for...
Nonetheless, the multiple-bit mechanical adder/subtractor is at heart pretty simple. I only installed five bits, because i only cared about the basic function:
5-bit adder/subtractor, no support for negative numbers. No extraneous output providers are needed, and fully optimised (i wasted one column to the right) it should require, per bit with carry input, 9x2 tiles, 31 power and 22 mechanisms for gears, connections, the rollers and the carry pressure plate (needed as input for the sum calculation, thus non-optional).
To the left, a bunch of even/odd gates taking input from lines A, B and the Carry for the current bit. This simply outputs the sum, as per usual, and functions the same in addition and subtraction.
Logically, it might look something like:
O☼P
P = power, O = Output, ☼ = AXBXC
To the right, the carry calculator.
Logically:
PABCO
☼
PABCO
☼
PABCO
P = power, A = on/on input from line A, B = on/on input from line B, C = Carry, O = output (sent to the sum calculation), ☼ = A XOR B connected gear. The power on/off on the carry of each cells serves as input for the next cell, but because of the difference gear there will be no power backflow from the next carry if the next cell's A and B are both on.
The whole contraption will also perform subtractions if you invert all A-Line connections for the carry calculations. In this simple implementation, it cannot handle negative numbers, though.
At least that's what i think it does. The basic function test was passed, but i can't exclude bugs or logical errors. The calculator doesn't need to be switched off to change input, or even to switch between addition and subtraction. Power connections just automatically reconfigure themselves. Carry calculations are near-instantaneous, because they work entirely through power flow without any signals apart from the main input. The sum signals can show a false positive for ~100 steps because of the unavoidable delay between a carry 'calculating' and sending its signal to the sum gear. But what's visible after ~120 steps should be the correct result.
[1] Someone suggested a construction depending 100% on gears to generate all output, without power-to-signal converters. I wasn't going to try that, the designs presented were absurdly oversized and -complicated at four bits already.