A Sharp Internal Congruence Modulo 2048 for the Overpartition Function

We establish a sharp internal congruence modulo 2048 for the overpartition function. Specifically, we prove that the number of overpartitions of 16n is congruent modulo 2048 to the number of overpartitions of 4n for every positive integer n. The proof uses a finite reduction of the overpartition generating function to representation numbers for sums of squares, followed by parity decomposition and theta-function dissections. We also show that the modulus 2048 is best possible.