Recognition quotients have exact logical prices

A counter can merge different event counts into the same observation. A repeated pair and exact comparison decisions determine where its cycle starts and how long it is. The smallest sufficient comparison window is determined exactly, with pairs of indistinguishable counters proving its necessity. Prime-factor and binary threshold searches recover the cycle with logarithmically many comparison calls; a decision-tree lower bound is attained for period-one counters and further families. Over an intuitionistic base, replacing the repeated pair by negative information makes general decidable classification equivalent to Markov's principle. Removing comparison decisions makes classification equivalent to excluded middle, even with a fixed repeated pair. These logical prices are joint results of Washburn and Zlatanovic. The finite-window and query results distinguish the information supplied by a collision from the calls needed to recover its index and period.