feat: show a conversation its own background tasks, and give it back every outcome

An `execute_task mode="async"` was invisible from the chat that started it.
The only trace was the receipt in the transcript and a row on the Tasks page
— which does not say *which* of those rows the assistant just spawned — so
"is it still going?" had no answer where the question is asked.

Worse, a task that did not simply succeed never came back at all. `run_job`
branched on `Ok`/`Err` first and routed by `job.kind` only inside the `Ok`
arm, so a failure or a kill left through the `Err` arm's unconditional
`hub.notify` — the home source (`/sethome`), worded "Cron job … failed" —
while the parent conversation sat waiting for a `task_completed` that would
never arrive. The wrong chat, and a wedged one.

The fix is a shape, not a branch: one `JobOutcome` classification, then one
`match job.kind` delivery site for every ending. An async task now ends in
its parent conversation whatever happened to it. The sink has a single
channel deliberately — to the model reading it, "it broke" is a result like
any other and must not be overlookable — so a failure is delivered as prose,
carrying whatever partial output the run produced, which is usually the only
clue about why. A cron job keeps the home notification: it belongs to nobody's
conversation. Cancellation becomes a third outcome rather than a flavour of
failure (`job_runs.status` has always had `'cancelled'` in its CHECK and
nothing ever wrote it), classified off the new typed `TurnCancelled` error so
nothing keys on a message string.

The strip above the composer is the visible half. `ServerEvent::TaskUpdate`
announces state to the source of the parent conversation only; the list is
`renderTaskStrip` (shared by the desktop copilot and the mobile chat), fed by
state on `ChatSession`. Each row links to `#session/{id}` — the page that
already shows, live, what a background agent is doing, and without which
"a task is running" is a fact you can do nothing with. Stopping is the
existing kill endpoint. A finished row clears itself after 20 s (its result
is in the conversation by then); a failed one stays until dismissed, and the
dismissal is remembered across reloads.

`GET /api/{source}/tasks` is what makes the strip survive a browser refresh:
the event is a broadcast with no replay, so without a load-time read a reload
would empty a chat that still has work running under it. It answers with the
running tasks plus failures from the last 30 minutes — the two states a person
can still act on. Successes are absent on purpose. Its window compares through
`datetime()` on both sides: `completed_at` is RFC 3339 and the cutoff is
SQLite-shaped, and `'T' > ' '` would let every same-day row through a window
meant to exclude it.

Not addressed, and worth doing next: a cron job's result should go where its
creator says, not always to the home chat.

crates/core-api/src/events.rs

@@ -285,6 +285,38 @@ pub enum ServerEvent {
     SecurityGroupSelected {
         group: String,
     },
+    /// A background task (`execute_task` with `mode: "async"`) started by this
+    /// conversation changed state.
+    ///
+    /// Emitted only for async tasks, and only to the source of the conversation
+    /// that started one: a cron job belongs to nobody's chat. It drives a live
+    /// view and nothing else — a client that misses it is merely out of date,
+    /// never out of sync, because the task's real ending is delivered into the
+    /// conversation's own history.
+    TaskUpdate {
+        job_id:   i64,
+        title:    String,
+        agent_id: String,
+        /// The task's own session — `#session/{id}` shows what it is doing.
+        session_id: Option<i64>,
+        state:    TaskState,
+        /// Why it ended badly. Set for `Failed` and `Cancelled`.
+        #[serde(skip_serializing_if = "Option::is_none")]
+        error:    Option<String>,
+    },
+}
+
+/// The lifecycle state of a background task in a [`ServerEvent::TaskUpdate`].
+/// Mirrors `job_runs.status`, plus the `Running` state that table only records
+/// by omission.
+#[derive(Clone, Copy, Serialize)]
+#[serde(rename_all = "snake_case")]
+pub enum TaskState {
+    Running,
+    Completed,
+    Failed,
+    /// Stopped by a human before it finished.
+    Cancelled,
 }
 
 impl ServerEvent {
@@ -324,6 +356,7 @@ impl ServerEvent {
             Self::TurnRunning        { .. } => "turn_running",
             Self::ClientSelected     { .. } => "client_selected",
             Self::SecurityGroupSelected { .. } => "security_group_selected",
+            Self::TaskUpdate         { .. } => "task_update",
         }
     }
 }