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.
collect ⇄ disperse
(A [B C]) = (A [B]) (A [C])
The simplifying direction (from right to left) is collect, and the expanding direction (from left to right) is disperse.
⇒ Frames
The generic structure of a frame is – generic frame:
(frame-type [frame-content])
The frame-type is Structure between round- and Square-Brackets. The generic template collapses if either the type or the content is Void-Equivalent. There are categories of frames that all have the same frame-type. In Arrangement collect gathers together framed-content that has in common the same frame-type. (p. 198)
🔺
BRICKEN, William, 2019. Iconic Arithmetic Volume I: The Design of Mathematics for Human Understanding. Unary press. ISBN 978-1-73248-513-6, p. 193–194, 198.
~
The room dialect maintains nesting through the availability of open doors that permit access to deeper rooms. Explicitly the outermost door leads outside.