This variety is very similar to the 1883 ODD-001 and 1898 ODD-001, both of which have die dots on the reverse near the center. Specimens with the cud are especially desirable. Since the discovery of the 1875 DOT-001, which was intentionally marked to catch a thief at the mint, it has been a mystery how these other die chips came to be. Two specimens from 1864 of different die states have been discovered that reveal how these dots could be made. Could they have been marked, too, by mint survivors to monitor shady employees?
No, it is simply die deterioration. For whatever reason, the reverse die tends to fail very near the center of the coin. The reason for the consistency is most likely the design. When a die fails on the interior, it commonly occurs at the same spot. For instance, the Goatees, 8th feature chip that is so common in 1899, 1892 Jeweled Forehead, and others.
Between these specimens of 1864, you can see how the die chip turns from a rather inconspicuous lump to a very well defined, square-shaped chip.