Lower bound on worst case pancake number?
I assume that you are talking about the Pancake sorting (wiki) problem. Your argument for $P_n \ge n$ is rather informal.

The paper "Bounds for Sorting by Prefix Reversal (1979)" gives the lower bound $P_n \ge n$ in the Introduction section. Better lower bounds are also mentioned in this paper.

This paper "Pancake Flipping Is Hard" (2011) proves that determining the minimal number of moves for a given stack is NP-hard.