To complete an Operational Description of Round- And Square-Brackets, we will need an axiom that specifies how the two boundaries interact in configurations more complex than the double nesting of Inversion. Here is the James axiom of Arrangement. The axiom specifies an invariant structure across frames.
In our chosen interpretation of exponential and logarithm, round- and Square-Brackets have an associated base that they implicitly share. The round-bracket ( ) contains exponents. The variable A in (A) is interpreted as an exponent while the round-bracket represents the arbitrary base, #A. Inversely, the square-bracket [ ] contains logarithmic content. The variable A in [A] is the form we are identifying the logarithm of, while the square-bracket represents the logarithmic function, log# A.