A `terminal(background=true, heartbeat=N)` tick queued a notification every N seconds
whether or not the process had printed anything, and every queued event costs the owning
session a full model turn. On Desktop and the TUI that turn painted the wake as a user
bubble ("[Background process ... heartbeat #9 ... (no new output since the last
heartbeat)]") followed by the model's "Still running normally." — over and over, for a
process whose row on the status stack already said it was running — and while the wake
held the session's turn, the user's own prompt sat queued behind it.
- `ProcessRegistry._emit_heartbeat` skips a tick with no new output. The sequence counts
delivered beats only; the "(no new output)" placeholder in the formatter is gone.
- TUI/Desktop type heartbeat rows `display_kind: hidden` (the kind both clients and the
transcript preview already honour); the CLI paints a one-line receipt and persists the
row hidden, so reopening the session in Desktop shows only the agent's reply.
- Desktop hydration drops heartbeat rows persisted by older backends the same way.
- `display.background_process_notifications: off` is honored by the TUI/Desktop poller and
the CLI drain, not just the messaging gateway. `off` mutes process-driven wakes only:
a finished `delegate_task(background=true)` still lands.
Supersedes #123123 (cherry-picked; scoped so `off` keeps subagent results) and #119202
(cherry-picked; `heartbeat: 0` is schema-valid so models that materialize every field
stop tripping the foreground guard).
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