The Minimax Discrimination Rate of General One-Dimensional Periodic Grid Codes

We study the minimax discrimination time of the full admissible class of one-dimensional periodic grid-cell codes with n binary neurons. For every integer n >= 1 and every separation 0 < rho <= 1/2 on the stimulus circle, the minimax time is infinite exactly when 0 < rho < 2^(-n). In the finite regime 2^(-n) <= rho <= 1/2, it satisfies (log 2/6) log_2(1/rho)/n <= T <= 64 log_2(1/rho)/n. The lower bound is obtained from a finite Abel-summation certificate uniform over the number of modules, neuron allocations, integer-nested scales, and receptive-field placements, while published extreme and balanced dyadic constructions provide the matching upper rate. The result is restricted to the one-dimensional integer-nested periodic model and does not assert an exact optimal constant, a single code simultaneously adaptive to every rho, a higher-dimensional theorem, or a non-integer scale-ratio result.