system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: What does BBC Radio 4's broadcasting status have to do with checks performed by a Trident CO before opening a letter of last resort?

Quote from Wikipedia: "According to Peter Hennessy's book The Secret State: Whitehall and the Cold War, 1945 to 1970, the process by which a Vanguard-class submarine commander would determine if the British government continues to function includes, amongst other checks, establishing whether BBC Radio 4 continues broadcasting."
gpt
Radio 4's long-wave broadcast is receivable over the entire planet, 24 hours a day, 365 days a year. Its not, obviously, receivable under the sea, but a submarine should be able to surface anywhere and pick up the signal.

Therefore if a British submarine surfaces and is unable to receive a Radio 4 broadcast, it's a reasonable assumption to make that something serious has happened to the UK (or at the very least the central London studios and broadcasting complex of the BBC). 

If the situation continues for several days, the submarine commander would be entitled to assume that there has been a catastrophic attack on critical UK infrastructure. He is then instructed to open the Letters of Last Resort which presumably authorise the commander to determine what's happened to Britain, and to take any revenge actions he deems necessary.

That's how the entire British nuclear-armed submarine deterrent is designed to work. The idea is that even if an enemy launched a surprise nuclear strike that wiped out the UK (which is after all a small island with most of the population centred in just a few very large cities), they know that somewhere out there under the ocean are submarines carrying multiple high-yield nuclear weapons. Sooner or later those submarines will surface near the coastline of whoever attacked Britain and launch their nuclear weapons in revenge.