fix(ui): poll for out-of-band email confirmation

An email link/code can be confirmed out of band: the recipient taps the
one-tap link in the email, which confirms in another browser/session. The
backend already publishes a `notify` `profile` re-fetch signal for this
(handlers_auth.go handleEmailConfirmLink), and the client re-fetches on it.

But the live stream is single-shot with no replay: a Mini App backgrounded
while the user is in their mail app tapping the link drops the stream and
misses the event (the gateway hub has no subscriber to deliver to), and the
reconnect on foreground does not re-sync — so the open code form stayed
until a manual reload. On Telegram Desktop the app is never backgrounded,
so the push works and there was no bug.

Add a client-side fallback: while an add-email confirmation is pending
(a code was sent, no email yet), poll `profile.get` on a 4s interval and on
foreground regain until the address lands; the effect stops as soon as the
email appears. The live push still updates instantly when foregrounded —
this only covers the backgrounded-miss gap.

Tests: a mock e2e attaches the email WITHOUT emitting a live event (new
window.__mock.clearEmail / confirmEmailOutOfBand seams), so it exercises the
poll, not the push, and asserts the code form collapses into the email row.
Docs: ARCHITECTURE.md §10 notes the single-shot gap + the poll fallback.

ui/e2e/social.spec.ts

@@ -327,6 +327,36 @@ test('change email: a free address replaces the current one', async ({ page }) =
   await expect(page.getByText('you@example.com')).toHaveCount(0);
 });
 
+// The add-email confirmation can complete out of band: the recipient taps the one-tap link in the
+// email, which confirms in another browser/session. A backgrounded Mini App misses the live
+// 'profile' event (the stream is single-shot, no replay), so the open code form falls back to
+// polling the profile until the address lands. The mock attaches the email WITHOUT emitting an
+// event, so only the poll can surface it.
+test('add email: an out-of-band confirmation surfaces on the open form via polling', async ({ page }) => {
+  await loginLobby(page);
+  await openProfile(page);
+
+  // Drop the seeded email so the sign-in section offers the add-email flow.
+  await page.evaluate(() => (window as unknown as { __mock: { clearEmail(): void } }).__mock.clearEmail());
+  const emailInput = page.locator('.accounts input[type="email"]');
+  await expect(emailInput).toBeVisible();
+
+  await emailInput.fill('linked@example.com');
+  await page.getByRole('button', { name: 'Send code' }).click();
+  await expect(page.locator('.accounts .codein')).toBeVisible();
+
+  // Confirmed elsewhere via the one-tap link, with no live event delivered: only the poll surfaces it.
+  await page.evaluate(() =>
+    (
+      window as unknown as { __mock: { confirmEmailOutOfBand(email: string): void } }
+    ).__mock.confirmEmailOutOfBand('linked@example.com'),
+  );
+
+  // The confirmed address surfaces as the email row, and the code form is gone.
+  await expect(page.locator('.acctrow').filter({ hasText: 'linked@example.com' })).toBeVisible({ timeout: 10000 });
+  await expect(page.locator('.accounts .codein')).toHaveCount(0);
+});
+
 test('link then unlink Telegram from the sign-in methods', async ({ page }) => {
   await loginLobby(page);
   await openProfile(page);