The phrase "after the break" is sometimes used before a commercial. It can also used in an article. e.g. "watch the video after the break".
Are these two usages related to each other, apart from having the same three words in the same order?
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The basic idea that is being communicated is the same — an interruption in continuity. The context, however, slightly changes the usage of the term. See the OED entry on 'break' (noun):
Break as used on television is definition 8k, and most often used in the context of a commercial/advertisement interruption.
Break as used in articles is definition 8c; it is used these days in the context of an in-text advertisement that the reader must scroll past to read the rest of the article, but it can also denote any other kind of break in continuity of the article — not just a 'commercial' interruption.