I'm not sure I follow. Multiplying through by two to get tau would simply move the nested square roots from the denominator to the numerator in the relation wit

I'm not sure I follow. Multiplying through by two to get tau would simply move the nested square roots from the denominator to the numerator in the relation with the golden ratio (as is explained in the link). I certainly don't see any step in the outlined derivation that gains or loses the ability to be intuited simply due to a factor of two.