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

In this framework, a completed recognition is required to remain distinguishable from its reversal; we formulate this requirement with a ledger of binary state changes.For a closed run, the primitive debit and credit increments balance; reversal therefore leaves the same state-account record. A phase account can retain the missing sign. When its domain is the complement of a recognized loop with process dimension of one, nonzero winding is equivalent to the homological requirement of the companion paper, and that requirement holds exactly when D=3. The account-domain restriction is an explicit premise; an ordered posting history can retain reversal information outside this restriction. The phase domain and recognized ownership of the registration are explicit physical premises. We also compare cubical processes whose recognized and registering trajectories are spheres of the same dimension k. Arity here defines the number of states that one act runs over, and not the number of arguments of an operation. Under the stated ambient homology hypotheses, their recognition substrate has a dimension of 2k+1. With additive tally costs and a sequential ledger, a loop costs exactly eight units to traverse or nine when including a unit phase account. An explicit closed schedule attains a surface minimum of ninety primitive units; every higher process costs more than the loop. If recognition takes the cheapest complete pass among these admissible process dimensions, that pass is the loop, and the substrate dimension is then three. Identifying the recognition substrate with the space carrying these acts gives the corresponding physical dimensional selection. The higher-dimensional alternatives are constructed.