Next we count the number of finite-length bars. Recall that each function g i is a Morse function. By Lemma A.1, the sum ∑ n i=1 g i (x i ) of Morse functions over a product space is also a Morse function. Hence V is Morse and as such, the birth and death times of the bars correspond directly to the critical points of V . Since 2 n of those points correspond to the birth times of the semi-infinite bars, we have ∏ n i=1 p i -2 n critical points remaining to be split equally as the birth and death times of the finite bars. It is important to note that the number of critical points of a Morse function on the circle will always be even: for every minimum that is introduced, a maximum is created since the Euler characteristic of the circle is zero (see also the Morse inequalities, which can be found on page 74 of Ref. 36). Thus the value ∏ n i=1 p i -2 n is even, which gives us 1 The same proof technique gives the following more general theorem.