I actually disagree with that logic as well. If you just knew that e^(i * pi) = -1, then how would you know, whether halving that exponent would yield +i or -i?

I actually disagree with that logic as well. If you just knew that e^(i * pi) = -1, then how would you know, whether halving that exponent would yield +i or -i? Considering that, we'd need to state Euler's identity as: e^(i * pi/2) = i, but again we'd know nothing about fractions of that exponent. And there we go! The point is, Euler's identity is a nice property of the definition of complex numbers, but in itself, not too generic. That's why OP's form, e^(i*tau) = 1, would do just fine. Edit: fixed asterisks.