The Process Dimension of Recognition: Arity, Cost, and the Persistence of the Completed Act

A companion paper shows that the space in which a recognizing system’s states sit has three dimensions if one requirement holds: after the system completes a full run through its states, the run must be kept by a second loop locked to the path of the first, and two loops lock only in three dimensions. That paper takes the requirement as a premise and takes for granted that a run is a loop. This paper supplies both. First, the requirement is shown to be the same thing as a plainer demand: that a completed run stay distinguishable from its own undoing. The system’s own books cannot hold that difference: a closed run and its undoing post the same entries to opposite columns, and on a closed run the two columns are equal, so the books read the same after either. If the difference survives at all, an account kept against a path that recognition has run holds it, and what such an account can hold is exactly the locked loop. Reading the record of the run as such an account, kept against a recognized path rather than against an arbitrary region, is the one identification the equivalence uses, and it is named where it is used. Second, a run could in principle sweep a surface rather than a loop, which would make the count 2k + 1 for a k-dimensional run. Every such alternative exists, and the cheapest one is the loop: eight units against a floor of thirty-one. The cheapest recognizer therefore runs a loop and lives in three dimensions.