Theorem 3. Suppose M 1 . . . M n are compact manifolds without boundary. Let m i denote the sum of the ranks of the homology groups of M i in all dimensions. Suppose each g i : M i → R is a Morse function with p i critical points. Note p i ≥ m i , and p i is odd if and only if m i is. Let M = ∏ n i=1 M i . Define the additive energy function Note, this theorem is a generalization of Theorem 2 in Ref. 6, generalizing from the n-alkane molecules (which are arranged in a linear chain) to now govern the sublevelset persistent homology of all branched alkanes, which is a broader family of molecules. Furthermore, instead of performing induction on the dimension of homology k as in Ref. 6, we use induction on the number of rotational axes n in order to prove our result. We consider induction on n as a way to reinforce the view that energy landscapes are nested functions. For example, in 2-methylpentane, the molecule with building block energy functions 1-3-2-1 and 1-2-2-1, the energy landscapes of both types can be found by slicing the energy landscape of 2-methylpentane parallel to the axes of the domain (Figure ). More generally, this inductive proof shows how one can use an understanding of the PEL of a smaller molecule in order to build up towards understanding the PELs of larger molecules.