Should be simple, I thought! Just a little baby step: hook the A and B register inputs in to the two 4-bit adders. Leave the complicated stuff for another lunch.
So: bus A to the adders' A inputs. Bus B to the adders B inputs. The Octal bus driver (one of the '245s I bought) is already connected to the bus; just need to wire up the Σ outputs to the '245. Aaand... nothing. Nothing? And why is one of the bits in register A suddenly being pulled to ground when I wire up this unrelated B register input to the other adder? Oh, and look! It's getting pretty hot...
Yank power. Quickly.
Oh, right. The adder on the left is a 74LS283, which I just bought because I only had one adder. The one that I had is a 74LS83. Which is an entirely different beast, with a different pinout.
Fortunately for me, I didn't destroy it by running power and ground to the wrong pins. And then I also didn't destroy it by tying ground to +5v and Vcc to GND, because the ground pin is on the north side of the IC and Vcc is on the south side, the opposite of normal these days.
Only took me about a half hour to diagnose what the heck I wired wrong and re-wire it badly. I guess lunch tomorrow is going to involve some clean-up of those higgledy-piggledy green jumpers.
The math even checks out! Whoohoo!