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.

web/components/shared/agent-tasks.js

@@ -0,0 +1,98 @@
+import { html, nothing } from 'lit';
+import { t } from '../../lib/i18n.js';
+
+/**
+ * The background-task strip: what the agent handed off to run on its own.
+ *
+ * Rendered just above the composer, and only when there is something to say —
+ * an empty strip would be a permanent slice of chrome for an occasional event.
+ * It sits outside the message flow on purpose: a task started twenty messages
+ * ago scrolls away exactly when it matters most, and the transcript is a record
+ * of what was said, not a dashboard.
+ *
+ * State lives on the chat component (`ChatSession._tasks`), which owns the
+ * WebSocket the updates arrive on; this file only knows how it looks. Shared by
+ * the desktop copilot and the mobile chat page.
+ */
+
+const STATE_ICON = {
+  running:   'bi-arrow-repeat agent-task-spin',
+  completed: 'bi-check-circle-fill',
+  failed:    'bi-exclamation-triangle-fill',
+  cancelled: 'bi-slash-circle',
+};
+
+/** Elapsed time since an ISO timestamp, coarse on purpose (`12s`, `4m 03s`). */
+function elapsed(startedAt) {
+  if (!startedAt) return '';
+  const ms = Date.now() - new Date(startedAt).getTime();
+  if (!Number.isFinite(ms) || ms < 0) return '';
+  const s = Math.floor(ms / 1000);
+  if (s < 60) return `${s}s`;
+  const m = Math.floor(s / 60);
+  if (m < 60) return `${m}m ${String(s % 60).padStart(2, '0')}s`;
+  return `${Math.floor(m / 60)}h ${String(m % 60).padStart(2, '0')}m`;
+}
+
+function renderTask(host, task) {
+  const running = task.state === 'running';
+  // The task's own session page is the drill-in: it already shows, live, what a
+  // background agent is doing. Without it "a task is running" is a fact you can
+  // do nothing with.
+  const openable = task.session_id != null && host._canOpenTaskSession;
+
+  return html`
+    <div class="agent-task agent-task--${task.state}">
+      <i class="bi ${STATE_ICON[task.state] ?? 'bi-hourglass'} agent-task-icon"></i>
+
+      <button
+        class="agent-task-body"
+        ?disabled=${!openable}
+        title=${openable ? t('chat.tasks.open') : ''}
+        @click=${() => { if (openable) window.location.hash = `session/${task.session_id}`; }}
+      >
+        <span class="agent-task-title">${task.title}</span>
+        <span class="agent-task-meta">
+          ${task.agent_id}
+          ${running ? html`· ${elapsed(task.started_at)}` : nothing}
+          ${task.state === 'failed'    ? html`· ${t('chat.tasks.failed')}`    : nothing}
+          ${task.state === 'completed' ? html`· ${t('chat.tasks.completed')}` : nothing}
+          ${task.state === 'cancelled' ? html`· ${t('chat.tasks.cancelled')}` : nothing}
+        </span>
+        ${task.error && task.state === 'failed'
+          ? html`<span class="agent-task-error" title=${task.error}>${task.error}</span>`
+          : nothing}
+      </button>
+
+      ${running
+        ? html`<button class="agent-task-action" title=${t('chat.tasks.stop')}
+                       @click=${() => host._stopTask(task.job_id)}>
+                 <i class="bi bi-stop-fill"></i>
+               </button>`
+        : html`<button class="agent-task-action" title=${t('chat.tasks.dismiss')}
+                       @click=${() => host._dismissTask(task.job_id)}>
+                 <i class="bi bi-x"></i>
+               </button>`}
+    </div>
+  `;
+}
+
+export function renderTaskStrip(host) {
+  const tasks = host._tasks ?? [];
+  if (tasks.length === 0) return nothing;
+
+  const running = tasks.filter(x => x.state === 'running').length;
+
+  return html`
+    <div class="agent-tasks">
+      <div class="agent-tasks-head">
+        <i class="bi bi-cpu"></i>
+        <span>${running > 0
+          ? t('chat.tasks.running_n', { n: running })
+          : t('chat.tasks.title')}</span>
+        <a class="agent-tasks-all" href="#tasks">${t('chat.tasks.see_all')}</a>
+      </div>
+      ${tasks.map(task => renderTask(host, task))}
+    </div>
+  `;
+}

web/components/shared/chat-page.js

@@ -2,6 +2,7 @@ import { html, nothing } from 'lit';
 import { ChatSession }   from '../../lib/chat-session.js';
 import { t }             from '../../lib/i18n.js';
 import { renderMsg, renderAttachmentChips } from '../copilot-render.js';
+import { renderTaskStrip } from './agent-tasks.js';
 
 export class ChatPage extends ChatSession {
   static properties = {
@@ -193,6 +194,7 @@ export class ChatPage extends ChatSession {
         </div>
 
         <div class="chat-page-input-area">
+          ${renderTaskStrip(this)}
           ${this._renderNoModelsBanner()}
           <div class="chat-page-composer"
                @dragover=${(e) => e.preventDefault()}