fix(watching): tell the page when the badge goes away

Reported from a manual test: a Codex session went on showing the watching badge
after its background terminal had finished. The server was right and the page
was stale — `Session.watching` changes while the session's status does not, and
nothing broadcast it.

The label is usually SET on the idle transition, which broadcasts anyway, so the
badge always appeared correctly. It CLEARS when the work ends, and a CLI can end
background work without taking a turn: codex repaints its background-terminal
row away and stays idle, so `_confirmIdle()` concludes without emitting `idle`
(that emit is guarded by `wasWorking || isInitialReady`) and no other event
fires. Every open page kept drawing a badge the server had already dropped.

`_readWatching()` now emits `watchingChanged` when, and only when, the label
really changes, and the wiring pushes the session state on it. No new SSE event:
the badge reads off the session payload every surface already has.

A/B measured on an isolated beta with the page loaded and then left untouched.
Without this commit the server dropped the label at t+50s and the page still
showed the badge at t+100s; with it, page and server cleared in the same
ten-second window.

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

src/session.ts

@@ -2763,7 +2763,17 @@ export class Session extends EventEmitter {
     // Called only from the probe, and only with what a capture returned: `null` is
     // "the screen could not be read", which is not evidence that nothing is running.
     if (!pattern || paneText === null) return;
-    this._watching = watchingLabel(paneText, pattern, this._watchingWindow);
+    const label = watchingLabel(paneText, pattern, this._watchingWindow);
+    if (label === this._watching) return;
+    this._watching = label;
+    // ⚠️ This CHANGES while the session's status does not, so it needs an event of its
+    // own. The label is usually set on the idle transition, which broadcasts anyway, but
+    // it CLEARS when the work ends — and for a CLI whose background work ends without
+    // taking a turn (measured on codex: a background terminal finishing repaints the row
+    // away and nothing else happens) the session is idle before and after. Without this,
+    // the server knew the badge was gone and every open page went on drawing it until
+    // some unrelated event arrived.
+    this.emit('watchingChanged');
   }
 
   /**

src/web/session-listener-wiring.ts

@@ -43,6 +43,7 @@ export interface SessionListenerRefs {
   exit: (code: number | null) => void;
   working: () => void;
   idle: () => void;
+  watchingChanged: () => void;
   taskCreated: (task: BackgroundTask) => void;
   taskUpdated: (task: BackgroundTask) => void;
   taskCompleted: (task: BackgroundTask) => void;
@@ -263,6 +264,17 @@ export function createSessionListeners(session: Session, deps: SessionListenerDe
       }
     },
 
+    /**
+     * Pushes the session state when `Session.watching` changes without the status
+     * changing with it. That is the badge appearing or, more often, going away: a CLI can
+     * finish its background work without taking a turn, so the row is idle before and
+     * after and no other broadcast fires. There is no SSE event of its own, because the
+     * badge reads off the session payload every surface already has.
+     */
+    watchingChanged: () => {
+      deps.broadcastSessionStateDebounced(session.id);
+    },
+
     // ─── Background Task Events ──────────────────────────────
 
     /** Broadcasts `task:created` — new background task discovered */
@@ -495,6 +507,7 @@ export function attachSessionListeners(session: Session, refs: SessionListenerRe
   session.on('exit', refs.exit);
   session.on('working', refs.working);
   session.on('idle', refs.idle);
+  session.on('watchingChanged', refs.watchingChanged);
   session.on('taskCreated', refs.taskCreated);
   session.on('taskUpdated', refs.taskUpdated);
   session.on('taskCompleted', refs.taskCompleted);
@@ -531,6 +544,7 @@ export function detachSessionListeners(session: Session, refs: SessionListenerRe
   session.off('exit', refs.exit);
   session.off('working', refs.working);
   session.off('idle', refs.idle);
+  session.off('watchingChanged', refs.watchingChanged);
   session.off('taskCreated', refs.taskCreated);
   session.off('taskUpdated', refs.taskUpdated);
   session.off('taskCompleted', refs.taskCompleted);