Assembly theory bridges the gap between evolutionary biology and physics by providing a framework to quantify the generation and selection of novelty in biological systems. We formalize the assembly space as an acyclic digraph of strings with 2-in-regular assembly steps vertices and provide a novel definition of the assembly index. In particular, we show that the upper bound of the assembly index depends quantitatively on the number b of unit-length strings, and the longest length N of a string that has the assembly index of N − k is given by N(N−1) = b2 + b + 1 and by N(N−k) = b2 + b + 2k for 2 ≤ k ≤ 9. We also provide particular forms of such maximum assembly index strings. For k = 1, such odd-length strings are nearly balanced. We also show that each k copies of an n-plet contained in a string decrease its assembly index at least by k(n − 1) − a, where a is the assembly index of this n-plet. We show that the minimum assembly depth satisfies d min(N) = ⌈log2(N)⌉, for all b, and is the assembly depth of a maximum assembly index string. We also provide the general formula for the lengths of the minimum assembly index strings having only one independent assembly step in their assembly spaces. Since these results are also valid for b = 1, assembly theory subsumes information theory.
