Disambiguating variable composition in description-complexity degrees at higher meta-levels

Develop a principled and canonical rule to disambiguate which variable to compose into when defining the non-commutative monoid operation on degree polynomials intended to capture description complexity across higher meta-levels of string machines.

Background

To relate description complexity to runtime, the authors propose assigning degree polynomials to transducers, with monoid operation given by composition (non-commutative). Extending this scheme to state categories introduces degrees of the form ay + bx + c and, at higher meta-levels, additional terms.

At higher meta-levels, it becomes unclear how to systematically determine which variable should be the target of composition in the degree expressions, creating an ambiguity that must be resolved to complete the proposed description-complexity framework.