Round Peg in a Square Hole

So, when I was a small eel I had a bunch of tests including the classic board with a bunch of shaped holes that have corresponding pegs that go in each hole. Well, I was always quite good at spacial relations, and always a smart ass. So, I finished all but the round and square pegs in a better time than most adults would get...
then spent about a minute "trying" to get the round peg into the square hole.

Cut to 30 years later, and my old thermostat is on the fritz

I wanted replace it with a fancy new Nest thermostat:

But it does not work with my old millivolt wall furnace.
So, I wired a new outlet (yes Mr. Inspector, I did put a new circuit in for it) and attached a 110V to 24V transformer:

Then sent both prongs of the 24V up to the thermostat (Rh and C), and sent the heater control wire back down (W1).

When the thermostat decides to turn the heat on, it does it by shorting Rh to W1, so I wired a relay to W1 and C (C on the transformer side)

Then wire the other side of the relay to the heater

Velcro it up to the wall

"Tidy" the whole mess up

Plug the thermostat into the socket:

And fire it up

Later I will tell you how it worked out with the poor lady administering the test, but that will require you ply me with bourbon first.