There some ideas here that have not gelled but I think they have some value.

I propose to dispose of the EQ paradox as follows:

A chain of objects is a sequence where for adjacent objects, a capability to one is held by the other. The holder is not required to be willing to use or pass or invoke that capability. A loop is a chain where the first object is the same as the last. In the scenarios we have seen so far, any loop that is constructible has nested incidence on each link.

Definition of nested incidence: As I proceed around the loop I traverse links. As I reach a new link I assign the link a new style of left parenthesis I record that parenthesis. If I traverse an old link I record the assigned form of parenthesis. I use the left form if I am traversing the link in the same direction and use the right form if I traverse the link in the direction opposite from which I first traversed the link. When I reach the starting point of the loop I will have recorded a set of parenthesis nested in the ordinary linguistic sense. For instance “[{]}” is not nested but “{[]}” is.

Disturbingly “({)(})” is also nested. The pattern is possible if there is a link with two ends at the same site. A loop can always be shrunk to length one by deleting stub ends. A stub end is a sequence that goes over a link from site p to site q and then directly back over the same link from q to p. In our word game a valid word can always be shrunk by repeatedly deleting two adjacent letters that are inverse to each other.

These ideas are probably connected with the mathematical free groups and the string of parenthesis are “words” as in the word problem. The right form of a paren is the inverse of the left form.

Counter to the above (sorry for the argument here!)

In free groups any element can come next in the word. Here only the links attached to the node we are at can come next. Perhaps this is more like category theory. But in category theory there are multiple arrows (links) between a given pair of objects. In the comm world multiple links are possible but the exception. Category arrows compose just as comm links compose. Arrows may or may not have inverses. All comm links have inverses. I think category theory doesn’t fit.

Rather than try to prove the assertion above let me record the insight that makes me believe it. It is the idea of exploring a maze without bread crumbs and without site EQ. As an explorer reaches a site he sees a set of outgoing links, all indistinguishable except for the one that he arrived on, to which he retains a capability.