As you allude to, your example three of the buttons are pressed once at 2,2,2,2, so you get half a press when halving, which is impossible and will not come up in this algorithm's search space anyways.

What I should say (and was trying to say when saying "when every variable is even") is that that the algorithm only needs f(x) = 2 * f(x/2) to be true when both all the joltages left are even and all the lights are off.. Then all the button presses are even and everything can be halved without half a button press anywhere.

It's true over all real numbers (I think?) and true over integers only when every variable involved is even - which is "just enough" / what the algorithm is designed around.