Let V : X 1 × . . . × X n → R be an additive function over a product space, meaning there are a collection of building block energy functions g i : X i → R so that V is defined as V (x 1 , . . . , x n ) = ∑ n i=1 g i (x i ). Let the bars in the sublevelset persistent homology barcodes of g 1 , . . . , g n have lengths { r } m r=0 , where 0 = ∞ and the lengths are ordered greatest to least, i.e. r > r+1 for all 0 ≤ r < m. Let y i,r be the number of bars in the sublevelset persistence of g i with length r . One can visualize this two-parameter family of numbers as being arranged in a matrix Y lengths = {y i,q } i∈{1,n} r∈{0,m} . For integers s and r with 0 ≤ s, r ≤ m, we define By convention, count n (r, s) = 0 if r < s. The following theorem holds more generally for any tensor product filtration, but we state it here only for the sublevelset persistence of an additive function over a product space.