fix(unified-mailbox): single-source unified counters, unified id space, background push

The unified-section sidebar badges (per-role unified folders + cross-view
All mail/Unread/Starred) failed to count down when messages were deleted/moved/
read from the unified views, and failed to count up for incoming mail - while
the underlying per-account folder counters updated correctly. Root cause: the
badges were a separate counter representation, recomputed only by a fresh server
fetch, completely decoupled from the optimistically-patched mailbox lists.

Three coordinated changes:

V1 - single source of truth: derive `unifiedCounts`/`crossUnreadCount` as a pure
live projection of `mailboxes` + `accountMailboxes` (the lists every mutation
already patches and push refreshes), over the last-known unified scope. A store
subscription re-projects whenever those lists change, so optimistic deletes and
push refreshes flow into the badges with no server round trip and no
eventual-consistency snap-back.

V3 - unified id space: searchEmails/advancedSearchEmails now namespace shared/
delegated mailboxIds (`${ownerId}:${id}`) like getEmails already did. The
cross-account views browse via advancedSearchEmails, so shared emails there
previously carried bare owner ids; now every fetch path is consistent and
emailInMailbox hits the `ids[mailbox.id]` fast path (originalId branches kept as
a defensive fallback). resolveSourceFolderName matches `m.id` first (also fixes
a latent missing source-folder name for shared emails).

Background push: bind push notifications for every connected login, not just the
active one - background accounts now drive the unified counters by rebuilding the
unified scope on their state changes. handleStateChange also refreshes the
mailbox list on a Mailbox change for ANY changed account key, so delegated
shared-folder activity arriving via the active client updates counters too.

Tests: unified-badge live projection on delete; client-level namespacing for
searchEmails/advancedSearchEmails (shared vs own account).

stores/__tests__/email-store-unified-actions.test.ts

@@ -3,6 +3,7 @@ import { useEmailStore } from '../email-store';
 import { useAuthStore } from '../auth-store';
 import type { Email, Mailbox } from '@/lib/jmap/types';
 import type { IJMAPClient } from '@/lib/jmap/client-interface';
+import type { UnifiedAccountClient } from '@/lib/unified-mailbox';
 
 // Regression coverage for issue #281: single-email actions performed in the
 // unified inbox must be routed to the *email's own account* client, not the
@@ -98,6 +99,9 @@ describe('unified-view single-email action routing (#281)', () => {
       processingReadStatus: new Set(),
       selectedEmail: null,
       selectedEmailIds: new Set(),
+      unifiedScope: [],
+      unifiedCounts: [],
+      crossUnreadCount: 0,
       emails: [
         // Second direct login: sourceClientAccountId === sourceAccountId === 'account-b'.
         makeEmail({ id: 'email-b', accountId: 'account-b', sourceClientAccountId: 'account-b', sourceAccountId: 'account-b', keywords: {}, mailboxIds: { 'b-inbox': true } }),
@@ -216,6 +220,62 @@ describe('unified-view single-email action routing (#281)', () => {
     expect(shared.totalEmails).toBe(9);  // was 10
   });
 
+  it('decrements the unified-section badges when deleting from the unified view (live projection)', async () => {
+    // The unified-section badges (unifiedCounts / crossUnreadCount) must be a
+    // live projection of the per-account mailbox lists, NOT a stale server
+    // snapshot. Deleting a message in the unified view patches the folder's
+    // counter; the badge must follow in lockstep without a re-fetch.
+    const scope: UnifiedAccountClient[] = [
+      {
+        accountId: 'account-a', accountLabel: 'A', client: activeClient,
+        clientAccountId: 'account-a', jmapAccountId: 'account-a', isShared: false,
+        mailboxes: [makeMailbox({ id: 'a-inbox', role: 'inbox' })],
+      },
+      {
+        accountId: 'owner-x', accountLabel: 'Shared', client: activeClient,
+        clientAccountId: 'account-a', jmapAccountId: 'owner-x', isShared: true,
+        mailboxes: [makeMailbox({ id: 'owner-x:x-inbox', originalId: 'x-inbox', role: 'inbox', isShared: true, accountId: 'owner-x' })],
+      },
+    ];
+
+    useEmailStore.setState({
+      mailboxes: [
+        makeMailbox({ id: 'a-inbox', role: 'inbox', unreadEmails: 2, totalEmails: 5 }),
+        makeMailbox({
+          id: 'owner-x:x-inbox', originalId: 'x-inbox', name: 'Shared Inbox',
+          role: 'inbox', isShared: true, accountId: 'owner-x',
+          unreadEmails: 4, totalEmails: 10,
+        }),
+      ],
+      emails: [
+        makeEmail({
+          id: 'email-shared', accountId: 'owner-x',
+          sourceClientAccountId: 'account-a', sourceAccountId: 'owner-x',
+          keywords: {}, // unread
+          mailboxIds: { 'x-inbox': true }, // BARE owner id (not namespaced)
+        }),
+      ],
+      selectedEmailIds: new Set(['email-shared']),
+    });
+
+    // Seed the badges from the scope (also stores unifiedScope).
+    useEmailStore.getState().refreshUnifiedCounts(scope);
+    useEmailStore.getState().refreshCrossCounts(scope);
+
+    const before = useEmailStore.getState();
+    expect(before.unifiedCounts.find(c => c.role === 'inbox')).toMatchObject({ unreadEmails: 6, totalEmails: 15 });
+    expect(before.crossUnreadCount).toBe(6);
+
+    await useEmailStore.getState().batchDelete(activeClient, true);
+
+    const after = useEmailStore.getState();
+    // Underlying folder counter dropped...
+    expect(after.mailboxes.find(m => m.id === 'owner-x:x-inbox')!.unreadEmails).toBe(3);
+    // ...and the unified-section badges followed via the live projection.
+    expect(after.unifiedCounts.find(c => c.role === 'inbox')).toMatchObject({ unreadEmails: 5, totalEmails: 14 });
+    expect(after.crossUnreadCount).toBe(5);
+  });
+
   it('still uses the active/passed client outside unified view', async () => {
     useEmailStore.setState({
       isUnifiedView: false,