feat(session): say when a session is watching its own background work

An agent that arms a monitor, backgrounds a shell or hands a task to a
cloud session is told to end its turn. The pane then falls quiet, Claude
Code's idle_prompt notification arrives a minute later, and every surface
files the session under NEEDS YOU with nothing for a human to answer.

Claude states what it is still running on the last row of its screen
(`⏵⏵ bypass permissions on · 1 monitor · ← for agents`). That row is now
`capabilities.workDetect.watchingLine` in the CLI registry, guarded by
compileVersionRegex() like every other config regex, and the idle probe
reads it off the capture it already takes: `watchingLabel()` in
session-activity.ts searches the last five lines only, so a session that
PRINTS "1 monitor" is not mistaken for one running it.

The label lands on Session.watching and rides toLightDetailedState() out
to every surface. The phone overview, the desktop home rail and the rich
sidebar rows wear it as a `watching` badge in the accent colour, beside
the state pill and never in place of it: an agent can arm a monitor and
ask a question in the same breath, and only the pill says which.

Verified end to end against a throwaway session on an isolated beta
instance: the payload carried `watching: "1 monitor"` once the turn
ended, the badge rendered next to a yellow `waiting` pill, and both
cleared when the monitor died.

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>

test/home-sessions.test.ts

@@ -356,3 +356,50 @@ describe('home screens: one order, one numbering', () => {
     expect(branch).not.toContain('this.sessionOrder[idx]');
   });
 });
+
+describe('home sessions column: watching badge', () => {
+  it('carries the label off the session payload', () => {
+    const app = loadHomeSessionsApp({
+      sessions: sessionMap([{ id: 'watcher', watching: '1 monitor' }, { id: 'plain' }]),
+      sessionOrder: ['watcher', 'plain'],
+      cases: CASES,
+    });
+
+    const rows = Object.fromEntries(app.buildHomeSessionRows().map((r: any) => [r.id, r.watching]));
+    expect(rows).toEqual({ watcher: '1 monitor', plain: '' });
+  });
+
+  it("builds the badge with the phone overview, in the rail's own pill class", () => {
+    // Both home screens must word this badge identically, so the rail borrows the
+    // builder rather than writing a second one — the same arrangement it already has
+    // for state classification and for the stamp wording.
+    const app = loadHomeSessionsApp({
+      sessions: sessionMap([{ id: 'watcher', watching: '2 shells' }]),
+      sessionOrder: ['watcher'],
+      cases: CASES,
+    });
+
+    const row = app._buildHomeSessionRow(app.buildHomeSessionRows()[0]);
+    const badges = collect(row).filter((el: any) => String(el.className).includes('--watching'));
+    expect(badges).toHaveLength(1);
+    expect(badges[0].className).toBe('home-sessions-pill home-sessions-pill--watching');
+    expect(badges[0].textContent).toBe('watching');
+    expect(badges[0].title).toBe('Still running in the background: 2 shells');
+  });
+
+  it('draws no badge for a session running nothing in the background', () => {
+    const app = loadHomeSessionsApp({
+      sessions: sessionMap([{ id: 'plain' }]),
+      sessionOrder: ['plain'],
+      cases: CASES,
+    });
+
+    const row = app._buildHomeSessionRow(app.buildHomeSessionRows()[0]);
+    expect(collect(row).filter((el: any) => String(el.className).includes('--watching'))).toHaveLength(0);
+  });
+});
+
+/** Every node under one the builders made, the row itself included. */
+function collect(node: any): any[] {
+  return [node, ...(node.children || []).flatMap((child: any) => collect(child))];
+}