Final answer:
The Defense Security Service does not have security cognizance over military installations; this is the responsibility of the military or other government agencies. The statement in the question is false. Furthermore, under the First Reconstruction Act passed in 1867, the South was indeed divided into military districts, which is true.
Step-by-step explanation:
The Defense Security Service (DSS), a federal security agency of the United States, has the responsibility to oversee the protection of U.S. and foreign classified information in the hands of industry.
However, it does not have security cognizance over military installations; this responsibility typically resides with the military itself or other designated government agencies. Therefore, the statement that the Defense Security Service has security cognizance over cleared facilities on a military installation is false.
Regarding the historical aspect of the question, under the First Reconstruction Act passed in 1867, the Southern states of the U.S. were indeed divided into military districts to ensure order and facilitate reconstruction after the Civil War. This fact is true and reflects a significant period in United States history known as Reconstruction.