Correlation between Assembly Index and symmetry

Ascertain whether a correlation exists between the Assembly Index a(đ’Ș), defined as the minimal length of an optimal assembly addition chain for an object đ’Ș in an assembly space (S, ∘, BB), and quantitative symmetry measures of đ’Ș, such as the length of the longest repeated substring for strings or the order of the automorphism group for graphs, and characterize the nature and strength of this relationship across relevant classes of objects.

Background

The paper introduces Assembly Addition Chains as a generalization of classical Addition Chains to arbitrary sets equipped with a gluing operation and designated building blocks, formalized via the Assembly Multi-Magma and Assembly Space structures. The Assembly Index a(đ’Ș) measures the minimal number of gluing steps required to construct an object đ’Ș from building blocks.

After developing lower and upper bounds for the Assembly Index in several families (strings, colored connected graphs, colored polyominoes), the authors speculate about a potential link between assembly complexity and structural symmetry. They note that certain objects with repetitive or self-gluing constructions (e.g., monochromatic string-like objects) appear intuitively more symmetrical, motivating a conjectured correlation between a(đ’Ș) and symmetry metrics (e.g., repeated substrings for strings, automorphism group order for graphs).

The authors explicitly flag this as a conjecture and suggest it is worthy of further investigation, positioning it as an open direction that would connect complexity-theoretic properties with symmetry-based descriptors across different Assembly Spaces.

References

Since objects characterized by Optimal Assembly Addition Chains for which every object is obtained by gluing the predecessor object with itself, e.g. the monochromatic string-like objects, are intuitively expected to be more symmetrical than objects with the same size characterized by Optimal Assembly Addition Chains where every object is the gluing of two different predecessor objects, it is natural to conjecture that there must be a correlation between the Assembly Index of an object $\mathscr{O}$ and its symmetry , (e.g. the length of the longest repeated sub-string, the order of its automorphism group, etc) and checking this is worth of further investigation.

— Assembly Addition Chains (2512.18030 - Cronin et al., 19 Dec 2025) in Section: Outlook and Conclusions (final paragraph)