Final answer:
The rule that mandates the blackout of a network show from a distant station by cable operators when a local affiliate broadcasts the same program is known as syndex or syndication exclusivity, enforced by the FCC.
Step-by-step explanation:
The rule that requires cable operators to black out a network show from a distant station if the local affiliate plans to broadcast the same program is known as syndex or syndication exclusivity. This rule is established by the Federal Communications Commission (FCC) and is grounded in the idea of protecting the rights of local television stations. Affiliates, which are local stations that broadcast national network programming, have agreements with their networks that typically give priority to network programming over other content. These affiliates have the exclusive right to air certain programs in their designated market area, and if a cable operator receives a program from a distant station that is also scheduled to be broadcast by a local affiliate, the operator must not show the distant station's version of the program.
It should be noted that the FCC has been responsible for various broadcast standards, including those governing the prohibition against airing obscene programming at any time or indecent programming during certain hours, as well as regulating instances when censorship is allowed for broadcast content. Cable operators have more leeway compared to broadcast television as they often do not use public airwaves and provide services through subscription-based models.