1. Introduction

Assembly theory was formulated in 2017^[1] , introducing the concept of assembly index (initially called "pathway complexity") of an object as the smallest number of steps required to assemble this object from a set of basic building units, forming an initial assembly pool and from subunits that entered the assembly pool in previous assembly steps. The assembly index is, therefore, a measure of the complexity of the object, which is computable^[5] , unlike Kolmogorov complexity, for example, and captures the structural information about the object, unlike Shannon entropy. The theoretical background for the theory was researched^[5] based on directed multigraphs showing that the assembly index of an object is computable for all finite objects.

Consider two binary strings C = [01010101] and D = [00010111] and the initial assembly pool containing two bits 0 and 1. Both strings have the same length N = 8 and the same Shannon entropy H(C) = H(D) = log₂(2) = 1. However, the assembly index of the first string is a(C) = 3 (In step 1, assemble "01" and put it into the assembly pool, in step 2 assemble "01" with "01" taken from the assembly pool and put "0101" into the assembly pool, and in step 3 assemble "0101" assembled in the second step with "0101" taken from the assembly pool), while the assembly index of the second string is a(D) = 6, since only the substring "01" can be reused from the assembly pool^[8].

The theoretical background for the theory was researched^[5] based on directed multigraphs showing that the assembly index of an object is computable for all finite objects.

Lower bound on the assembly index - OEIS A003313 (red); log₂(𝑁) (red, dash–dot),

conjectured upper bound on the bitstring assembly index (green) and 𝑁−1 (green, dash–dot).

Basic building units depend on a particular application of the assembly theory. In chemistry, it found applications in drug discovery^[3]. Furthermore, the theoretical value of the assembly index of a molecule, where the initial assembly pool contains chemical bonds, can be experimentally confirmed using tandem mass spectrometry, nuclear magnetic resonance, or infrared spectroscopy^[4][7]. Therefore, the assembly index is the universal threshold between abiotic and biotic molecules and a robust and simple biosignature^[1][2] to distinguish random, abiotic objects from biologically or technologically assembled ones, as only biotic samples can have a molecular assembly index above 15. The more complex a given object, the less likely an identical copy can exist without some information-driven mechanism that generates that object^[6].