List a session's background tasks above its subagents

The count beside the status said how much work was going and never what,
so "3 bg tasks" was a number with no way to find out what it was about.

Drivers now report the tasks themselves rather than a size:
`Driver::background_tasks` returns `Vec<BackgroundTask>` -- id, the
provider's own description, and a kind -- served by
`GET /sessions/{id}/background`. It is runtime state, never persisted,
and `null` is "nobody has said", which is what a session with no process
answers and what the panel says in words rather than drawing as an empty
list. `description` is optional because Codex names a background terminal
by a process id, and a number drawn as a name is worse than admitting
there is none.

Claude's `background_tasks_changed` entries turn out to be objects
carrying `task_id`, `task_type` and `description`, so each is read rather
than counted -- and an `ambient` one is now dropped from the list and the
count alike, on the CLI's own instruction: a live-update watcher is not
activity, and counting one left a session reading `waiting` with nothing
to wait for.

The phone draws them in the right-hand panel above the subagents,
collapsed to "2 bg tasks running" and pushing the subagents down when
opened. Both lists are items of one lazy column, so neither can run off
the panel, and the section is refetched whenever the live count moves --
a card for work that has finished is exactly the stale measurement the
count exists not to be.

Verified against the real Claude CLI (2.1.261): a backgrounded `sleep 120`
came back as `{"id":"br16327wr","description":"Sleep for 120 seconds",
"kind":"command"}`, and on the emulator against the echo rig the section
appeared, expanded, and dropped a card as its task finished.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>

app/androidApp/src/main/kotlin/com/example/aiapp/Api.kt

@@ -308,6 +308,46 @@ data class SubagentSummary(
     val lastActivity: Double,
 )
 
+/**
+ * One thing a session has running while it is free to do something else: a backgrounded command, a
+ * subagent, whatever else its provider can leave going. See `GET /sessions/{id}/background`.
+ */
+data class BackgroundTaskSummary(
+    /** The provider's own id. Never shown -- it is what keys the list and matches two snapshots. */
+    val id: String,
+    /**
+     * What it is doing, in the provider's own words. Null where it names a task by nothing a reader
+     * would recognise -- Codex reports a process id -- and the card says so instead.
+     */
+    val description: String?,
+    /** `agent`, `command`, `workflow`, or `other` for a kind this build has not heard of. */
+    val kind: String,
+)
+
+/**
+ * What [sessionId] has running in the background right now.
+ *
+ * Null is "its provider has not said", which a stopped session and an old backend both answer, and
+ * is a different thing from the empty list.
+ */
+fun fetchBackgroundTasks(
+    settings: ServerSettings,
+    sessionId: String,
+): List<BackgroundTaskSummary>? =
+    requestFromServer(settings, "/sessions/$sessionId/background") { connection ->
+        val body = connection.inputStream.bufferedReader().readText()
+        if (body.trim() == "null") null
+        else
+            JSONArray(body).mapObjects { row ->
+                BackgroundTaskSummary(
+                    id = row.getString("id"),
+                    description =
+                        if (row.isNull("description")) null else row.getString("description"),
+                    kind = row.getString("kind"),
+                )
+            }
+    }
+
 fun fetchSubagents(settings: ServerSettings, sessionId: String): List<SubagentSummary> =
     requestFromServer(settings, "/sessions/$sessionId/subagents") {
         it.jsonObjects { row ->