Show live background task counts

server/src/session/driver.rs

@@ -337,6 +337,14 @@ pub enum Event {
         id: String,
         answers: Vec<String>,
     },
+    /// How many background tasks the provider says are alive now.
+    ///
+    /// State rather than a transcript row: the phone draws it beside the
+    /// session status. A distinct event keeps the count current while a
+    /// session is open; `GET /sessions` supplies the opening snapshot.
+    BackgroundTasks {
+        count: usize,
+    },
     Status {
         state: SessionStatus,
     },
@@ -596,6 +604,12 @@ pub type EventSink = mpsc::UnboundedSender<Event>;
 /// with live input injects it at the next tool boundary, while a turn-at-a-time
 /// dialect queues it for the next child process.
 pub trait Driver: Send + Sync {
+    /// The provider's latest measured number of live background tasks.
+    /// `None` means it has not reported one, not that the count is zero.
+    fn background_tasks(&self) -> Option<usize> {
+        None
+    }
+
     /// Takes a message, now or once the session is free for it.
     ///
     /// Every driver owes exactly one `MessageTaken` per message, at the moment