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>

src/web/public/app.js

@@ -4574,6 +4574,11 @@ class CodemanApp {
     return {
       state,
       pill: this._sidebarRichPillLabel(state),
+      // What the pane's own footer says is still running in the background ("1 monitor",
+      // "2 shells"). A row that has one went quiet because the agent is waiting for that,
+      // which is a different thing from waiting for the user — so it rides BESIDE the
+      // state pill and never replaces it.
+      watching: typeof session.watching === 'string' ? session.watching : '',
       createdAt: Number(session.createdAt) || 0,
       since: this._mobileOverviewSince ? this._mobileOverviewSince(state, session) : null,
     };
@@ -4609,6 +4614,13 @@ class CodemanApp {
       parts.push(stamp(row.since.key, row.since.at, 'for', 'tab-meta-since'));
     }
     parts.push(`<span class="tab-pill tab-pill--${escapeHtml(row.state)}">${escapeHtml(row.pill)}</span>`);
+    // The word is duplicated from mobile-overview.js for the same reason the pill labels
+    // above are: it is one word, and this file must render a complete row even when a
+    // stale cached mobile-overview.js has arrived without it.
+    if (row.watching) {
+      const title = escapeHtml(`Still running in the background: ${row.watching}`);
+      parts.push(`<span class="tab-pill tab-pill--watching" title="${title}">watching</span>`);
+    }
     // Both absolute stamps ALSO on the line itself, not only on the two items.
     // Below 288px the rail hides `.tab-meta-created` (the `tab-rail-tight`
     // rule), and a tooltip on a `display: none` element has no hover target —
@@ -4646,7 +4658,7 @@ class CodemanApp {
     const prev = tab.dataset.tabState;
     // The since ANCHOR moves without the state changing (each new turn re-stamps
     // lastSubmitAt), so key the compare on both.
-    const sig = `${row.state}:${row.since ? row.since.at : 0}:${row.createdAt}`;
+    const sig = `${row.state}:${row.since ? row.since.at : 0}:${row.createdAt}:${row.watching}`;
     if (tab.dataset.tabMetaSig === sig) return;
     tab.dataset.tabMetaSig = sig;
     tab.dataset.tabState = row.state;