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

A completed recognition should remain distinguishable from its reversal. We formulate this demand using 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 trajectory, nonzero winding is equivalent to the homological requirement that selects three dimensions in the companion paper. The account-domain restriction is an explicit premise; an ordered posting history can retain reversal information outside this restriction. We also compare cubical processes whose recognized and registering trajectories are spheres of the same dimension k. Under the stated ambient homology hypotheses their substrate dimension is 2k+1. With additive tally costs and a sequential ledger, a loop costs exactly eight units to traverse, or nine including a unit phase account. An explicit closed schedule attains the surface minimum of ninety primitive units; every higher process costs more than the loop. Least cost therefore selects loops within this admissible class. The higher-dimensional alternatives are constructed.