Picked up
A helper still working when the lead's turn ends keeps going, and so does a command the lead started to wake the discussion. When one finishes, the discussion is picked up: the lead gets one short turn with what came back, under the line Picked Up When the Background Task Finished, with the report folded beneath it, and you get a notice. Several finishing within a few seconds of each other are one turn. That turn has nobody to ask: it asks you no questions and runs nothing that needs your approval, and a helper's changes stay on its branch for you to review — it sets them out, it never merges them.
When it is not
It is not picked up when you pressed Stop (which also stops helpers and watchers that outlived the run), when the discussion has reached a spending ceiling, or when your team's plan has lapsed. To keep a lead from waking itself for ever, a discussion is picked up at most three times in a row without a word from you, six times an hour, and a minute apart. When one of those holds it, the line reads Background Task Finished, Not Picked Up, says why, and keeps the report; write in the discussion to carry on.
Turning it off for one discussion
Don't Pick Up Background Results in the discussion's ⋯ menu — on the Mac, the web and the iPhone, a checkmark when on — keeps that discussion from being picked up. What finishes there gets the line Background Task Finished, Not Picked Up — Turned Off for This Discussion, with its report folded under it; watchers keep running and keep showing under Background Tasks. It belongs to the discussion and travels with it, not to Settings.
A phone following the discussion says so with a notification — the discussion's name and "picked up a finished background task" — under the same switch as When a Discussion Finishes. Tapping it opens the discussion.
Watchers
The lead can set a watcher instead of checking something again and again: a pull request's checks finishing, a pull request becoming mergeable, a command succeeding or printing a given text, an address answering, a file appearing, another of your discussions on this Mac posting its next turn or finishing its run. A watcher checks every 30 seconds at the most often, never calls a model, and always gives up — after four hours at the most. When it sees what it waits for, or gives up, the discussion is picked up as above, with the instruction the lead left itself. A discussion runs at most three at once. They show under Background Tasks as Watching, Met or Gave Up, with Stop, here and on the web and the iPhone. A watcher's command is asked about once, like any command, and runs under the same sandbox. A watcher on a discussion hands the lead what that discussion said, with secrets masked and marked as another room's words rather than yours; it wakes the lead once, so a lead following every turn starts another each time — and the limits above still apply. It only watches discussions the lead could open, never its own. Watchers live on this Mac: one with time left carries on after the app restarts, and one whose time ran out while the app was closed is marked interrupted and the discussion says so.