So it turns out I absolutely hate wiring diode arrays, and I would like to display all three registers, not just the input register and accumulator. (I was able to rework things to get it back down to three registers). 

The seven segment LED display would require over 1000 diodes and I just don't care to do that.

So instead I've found a good deal on 24x IN-12B Nixie tubes and sockets and will use those instead. That eliminates the last few diodes and any semiconductors in the entire project.

In other news, I calculated that the core of the calculator, which can multiply two single digit numbers and then add another single digit number to the product, and then convert the binary result to BCD, will take 87 relays. It has a 34 relay latency that data has to propagate through. I measured the switching time of the relays I'm using in this part to be about 15ms, so it will take approximately half of a second for a signle digit calculation to take place.

This means that an addition or subtraction will take 4 seconds, a multiplication will take 4 seconds per multiplier digit (32 seconds for 8 digit x 8 digit) and a division, (which uses magnitude comparators, shifters, and counting subtraction) will take a maximum of 320 seconds (99999999/1) but will typically be much less.