feat(approvals): let a watching session keep quiet, and fix the tui gate

The badge alone left the row in NEEDS YOU, which is the thing the issue was
about. The fix is the alert that does not fire.

An idle prompt from a session that is watching its own background work now
opens ALREADY acknowledged. `hook-event-routes` passes `Session.watching` to
`notePrompt()`, which sets `acknowledgedAt` and records why in a new
`acknowledgedReason`. Nothing new suppresses anything: `acknowledge()` has
always meant "the alert this prompt armed is spent", and the prompt itself
stays pending, answerable and available as Read My Mind context. A wrong label
therefore costs a card that does not blink, never an alert that was never
created.

Every surface follows from that. The broadcast carries the reason, so a live
page declines to arm the tab alert and raises no desktop notification. The push
is skipped, since a false alarm is hardest to ignore on a phone. A reloading
page reads `acknowledgedAt` in `seedApprovals()`, which it already did. And
`classifySession()` now reads it too, which is a pre-existing bug fixed here:
acknowledging on one device cleared the alert everywhere except `codeman tui`.
It re-arms for free, because the next idle prompt supersedes the item and is
built fresh. Only `idle` is eligible, so a dialog that blocks the agent still
goes red whatever else it started.

The label is pane-derived and therefore prompt-injectable, so it is now read
from the last two rows of the screen only, with Claude's pattern anchored on
the `·` its footer joins items with, ANSI-stripped and length-capped at the
source. An agent that prints `· 1 monitor ·` into its own output finds no
match.

Verified on an isolated beta: a session that armed a monitor took its idle
prompt acknowledged with no alert on any surface, wore the badge, and showed
"quiet, watching 1 monitor" on its still-answerable card; the same session with
the monitor killed alerted normally on the next prompt. `test/watching-no-alert.test.ts`
pins both directions across all four surfaces.

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

src/web/routes/hook-event-routes.ts

@@ -176,6 +176,12 @@ export function registerHookEventRoutes(
     // identity beyond the shared per-instance secret, so a prompt claimed for a
     // session that can never show one must not create an answerable item).
     let approvalId: string | undefined;
+    // Set when the item opened ALREADY acknowledged, which today means the session is
+    // watching work it started itself. It rides the broadcast so a live page declines to
+    // arm the alert (a reloading page learns the same thing from `acknowledgedAt` when it
+    // seeds from /api/approvals), and it suppresses the push: an alert nobody can answer
+    // is worth even less on a phone than in a tab.
+    let acknowledgedReason: string | undefined;
     const approvalKind = APPROVAL_KIND_BY_EVENT[event];
     if (session && hooksAvailableForMode(session.mode, sessionHookOptions(session))) {
       if (approvalKind) {
@@ -194,6 +200,10 @@ export function registerHookEventRoutes(
           toolSummary: typeof toolSummary === 'string' ? toolSummary : undefined,
           message: typeof safeData.message === 'string' ? safeData.message : undefined,
           cwd: typeof safeData.cwd === 'string' ? safeData.cwd : undefined,
+          // What the pane says is still running in the background. An idle prompt from a
+          // session that is watching its own work opens acknowledged, so it never arms an
+          // alert nobody can answer; notePrompt() carries the whole reasoning.
+          watching: session.watching,
           // Visible tmux frame first (it IS the dialog); raw byte-buffer tail as
           // the fallback for direct-PTY sessions and the no-op test mux.
           capture: () => {
@@ -203,6 +213,7 @@ export function registerHookEventRoutes(
           },
         });
         approvalId = item.id;
+        acknowledgedReason = item.acknowledgedReason;
       } else if (APPROVAL_RESOLVING_EVENTS.has(event)) {
         approvalInbox.resolveForSession(sessionId, 'resolved_in_terminal');
       }
@@ -213,6 +224,7 @@ export function registerHookEventRoutes(
       timestamp: Date.now(),
       ...safeData,
       ...(approvalId && { approvalId }),
+      ...(acknowledgedReason && { acknowledgedReason }),
     });
     // Full state ride-along, same shape as the working/idle handlers: the home
     // screens rank the blocked group on lastActivityAt, and without this a
@@ -224,12 +236,17 @@ export function registerHookEventRoutes(
     // on approvalId, and the answer route refuses keystrokes for dsh dialogs
     // (third-party TUI, unmeasured contract) — so a dsh push stays a plain
     // notification instead of offering buttons whose answer would be refused.
-    ctx.sendPushNotifications(`hook:${event}`, {
-      sessionId,
-      sessionName,
-      ...safeData,
-      ...(approvalId && session?.mode !== 'deepseek' && { approvalId }),
-    });
+    // Nothing to push for a prompt that opened acknowledged: the agent is waiting for its
+    // own monitor or backgrounded shell, and a phone buzzing about it is the same false
+    // alarm as the tab alert, delivered where it is hardest to ignore.
+    if (!acknowledgedReason) {
+      ctx.sendPushNotifications(`hook:${event}`, {
+        sessionId,
+        sessionName,
+        ...safeData,
+        ...(approvalId && session?.mode !== 'deepseek' && { approvalId }),
+      });
+    }
 
     // Track in run summary. `prompt_submitted` fires on EVERY prompt of every
     // Claude pane; only the ones where the conversation actually moved (a /clear