Ah, yes. Generally you're right: on 2nd order functions ((nat -> nat) -> nat) TMs and LC differ where TM's have strictly more functionality. But then it flips a

Ah, yes. Generally you're right: on 2nd order functions ((nat -> nat) -> nat) TMs and LC differ where TM's have strictly more functionality. But then it flips again at 3rd order (((nat -> nat) -> nat) -> nat) and so on and so forth. Additionally, you can always just Gödel encode a program, pass it in, get all of this functionality. That might be considered cheating (simulating a TM, although you could just as well simulate LC in both models). This is not actually any different from what the UTM is doing... it's just possible to ignore the difference between interpretation and normal operation because it's all just tape.