None of this is true.
>fibonacci growth is strictly faster than exponential
There is even an explicit formula for F(n) that shows fibonacci growth is exactly exponential[1]
>In fact, this is why nlog(n) is the lower bound on the number of comparisons a comparison-based sorting algorithm
It's true that lower bound for comparison-based sorting is nlog(n) but it has nothing to do with fibonacci
>n! is approximately nlog(n)
Stirling's formula[2] says otherwise (that n! is exponential)
[1] http://en.wikipedia.org/wiki/Fibonacci_number#Closed-form_ex...