Minimax Discrimination and the Cost of Adaptation in Periodic Grid Codes

We study how a finite population of periodic binary neurons can distinguish positions at different spatial precisions. For the full one-dimensional integer-nested grid-cell class of Ost and Reynaud-Bouret, we determine the pointwise minimax discrimination time up to universal constants at every resolvable precision. We then ask how much is lost when one code must serve a whole range of precisions. Matching bounds determine this adaptation cost for every feasible population and integer precision budget, and thereby answer the two grid-code questions left open by Ost and Reynaud-Bouret. The loss is logarithmic in the number of dyadic precision levels when the population is at least twice the budget, but linear in the population when it equals the budget. The proof separates the neurons needed for resolution from those available to improve speed: a scale-resource identity and a bound on the number of response words give the lower bound, and a harmonic allocation attains the matching order. The orders persist when each receptive field has a fixed bounded number of interval components. For one interval per neuron, integer reciprocal periods without nesting allow constant-factor adaptation with sufficient redundancy. At critical precision, however, every resolving code with at least two neurons, nested or not, has dyadic periods and competitive cost equal to the population size. Thus period architecture and binary capacity impose different constraints on adaptation.